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G = C23×C33⋊C2  order 432 = 24·33

Direct product of C23 and C33⋊C2

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C23×C33⋊C2, C63⋊7C2, C62⋊36D6, C33⋊7C24, (C2×C62)⋊14S3, (C32×C6)⋊7C23, C32⋊9(S3×C23), (C3×C62)⋊19C22, C3⋊2(C23×C3⋊S3), C6⋊2(C22×C3⋊S3), (C3×C6)⋊9(C22×S3), (C22×C6)⋊5(C3⋊S3), (C2×C6)⋊12(C2×C3⋊S3), SmallGroup(432,774)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C23×C33⋊C2
Chief series
C1 — C3 — C32 — C33 — C33⋊C2 — C2×C33⋊C2 — C22×C33⋊C2 — C23×C33⋊C2
Lower central
C33 — C23×C33⋊C2
Upper central
C1 — C23

Generators and relations for C23×C33⋊C2
 G = < a,b,c,d,e,f,g | a2=b2=c2=d3=e3=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, gdg=d-1, ef=fe, geg=e-1, gfg=f-1 >

Subgroups: 9832 in 1876 conjugacy classes, 499 normal (5 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, C32, D6, C2×C6, C24, C3⋊S3, C3×C6, C22×S3, C22×C6, C33, C2×C3⋊S3, C62, S3×C23, C33⋊C2, C32×C6, C22×C3⋊S3, C2×C62, C2×C33⋊C2, C3×C62, C23×C3⋊S3, C22×C33⋊C2, C63, C23×C33⋊C2
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, C2×C3⋊S3, S3×C23, C33⋊C2, C22×C3⋊S3, C2×C33⋊C2, C23×C3⋊S3, C22×C33⋊C2, C23×C33⋊C2

Smallest permutation representation of C23×C33⋊C2
►On 216 points
Generators in S216
(1 167)(2 168)(3 166)(4 182)(5 183)(6 181)(7 169)(8 170)(9 171)(10 57)(11 55)(12 56)(13 175)(14 176)(15 177)(16 178)(17 179)(18 180)(19 66)(20 64)(21 65)(22 184)(23 185)(24 186)(25 187)(26 188)(27 189)(28 190)(29 191)(30 192)(31 193)(32 194)(33 195)(34 196)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(118 172)(119 173)(120 174)

(1 86)(2 87)(3 85)(4 101)(5 102)(6 100)(7 88)(8 89)(9 90)(10 138)(11 136)(12 137)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 147)(20 145)(21 146)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 118)(38 119)(39 120)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(64 91)(65 92)(66 93)(112 193)(113 194)(114 195)(115 196)(116 197)(117 198)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(139 166)(140 167)(141 168)(142 169)(143 170)(144 171)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(172 199)(173 200)(174 201)

(1 32)(2 33)(3 31)(4 47)(5 48)(6 46)(7 34)(8 35)(9 36)(10 192)(11 190)(12 191)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 201)(20 199)(21 200)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(37 64)(38 65)(39 66)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(91 118)(92 119)(93 120)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(145 172)(146 173)(147 174)(166 193)(167 194)(168 195)(169 196)(170 197)(171 198)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(181 208)(182 209)(183 210)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)(37 38 39)(40 41 42)(43 44 45)(46 47 48)(49 50 51)(52 53 54)(55 56 57)(58 59 60)(61 62 63)(64 65 66)(67 68 69)(70 71 72)(73 74 75)(76 77 78)(79 80 81)(82 83 84)(85 86 87)(88 89 90)(91 92 93)(94 95 96)(97 98 99)(100 101 102)(103 104 105)(106 107 108)(109 110 111)(112 113 114)(115 116 117)(118 119 120)(121 122 123)(124 125 126)(127 128 129)(130 131 132)(133 134 135)(136 137 138)(139 140 141)(142 143 144)(145 146 147)(148 149 150)(151 152 153)(154 155 156)(157 158 159)(160 161 162)(163 164 165)(166 167 168)(169 170 171)(172 173 174)(175 176 177)(178 179 180)(181 182 183)(184 185 186)(187 188 189)(190 191 192)(193 194 195)(196 197 198)(199 200 201)(202 203 204)(205 206 207)(208 209 210)(211 212 213)(214 215 216)

(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 216 19)(11 214 20)(12 215 21)(25 37 28)(26 38 29)(27 39 30)(31 48 40)(32 46 41)(33 47 42)(34 51 43)(35 49 44)(36 50 45)(52 64 55)(53 65 56)(54 66 57)(58 75 67)(59 73 68)(60 74 69)(61 78 70)(62 76 71)(63 77 72)(79 91 82)(80 92 83)(81 93 84)(85 102 94)(86 100 95)(87 101 96)(88 105 97)(89 103 98)(90 104 99)(106 118 109)(107 119 110)(108 120 111)(112 129 121)(113 127 122)(114 128 123)(115 132 124)(116 130 125)(117 131 126)(133 145 136)(134 146 137)(135 147 138)(139 156 148)(140 154 149)(141 155 150)(142 159 151)(143 157 152)(144 158 153)(160 172 163)(161 173 164)(162 174 165)(166 183 175)(167 181 176)(168 182 177)(169 186 178)(170 184 179)(171 185 180)(187 199 190)(188 200 191)(189 201 192)(193 210 202)(194 208 203)(195 209 204)(196 213 205)(197 211 206)(198 212 207)

(1 26 8)(2 27 9)(3 25 7)(4 39 23)(5 37 24)(6 38 22)(10 207 204)(11 205 202)(12 206 203)(13 28 16)(14 29 17)(15 30 18)(19 212 209)(20 213 210)(21 211 208)(31 52 34)(32 53 35)(33 54 36)(40 55 43)(41 56 44)(42 57 45)(46 65 49)(47 66 50)(48 64 51)(58 79 61)(59 80 62)(60 81 63)(67 82 70)(68 83 71)(69 84 72)(73 92 76)(74 93 77)(75 91 78)(85 106 88)(86 107 89)(87 108 90)(94 109 97)(95 110 98)(96 111 99)(100 119 103)(101 120 104)(102 118 105)(112 133 115)(113 134 116)(114 135 117)(121 136 124)(122 137 125)(123 138 126)(127 146 130)(128 147 131)(129 145 132)(139 160 142)(140 161 143)(141 162 144)(148 163 151)(149 164 152)(150 165 153)(154 173 157)(155 174 158)(156 172 159)(166 187 169)(167 188 170)(168 189 171)(175 190 178)(176 191 179)(177 192 180)(181 200 184)(182 201 185)(183 199 186)(193 214 196)(194 215 197)(195 216 198)

(1 86)(2 85)(3 87)(4 94)(5 96)(6 95)(7 108)(8 107)(9 106)(10 132)(11 131)(12 130)(13 101)(14 100)(15 102)(16 120)(17 119)(18 118)(19 124)(20 126)(21 125)(22 110)(23 109)(24 111)(25 90)(26 89)(27 88)(28 104)(29 103)(30 105)(31 60)(32 59)(33 58)(34 81)(35 80)(36 79)(37 99)(38 98)(39 97)(40 74)(41 73)(42 75)(43 93)(44 92)(45 91)(46 68)(47 67)(48 69)(49 83)(50 82)(51 84)(52 63)(53 62)(54 61)(55 77)(56 76)(57 78)(64 72)(65 71)(66 70)(112 195)(113 194)(114 193)(115 216)(116 215)(117 214)(121 209)(122 208)(123 210)(127 203)(128 202)(129 204)(133 198)(134 197)(135 196)(136 212)(137 211)(138 213)(139 168)(140 167)(141 166)(142 189)(143 188)(144 187)(145 207)(146 206)(147 205)(148 182)(149 181)(150 183)(151 201)(152 200)(153 199)(154 176)(155 175)(156 177)(157 191)(158 190)(159 192)(160 171)(161 170)(162 169)(163 185)(164 184)(165 186)(172 180)(173 179)(174 178)
 
G:=sub<Sym(216)| (1,167)(2,168)(3,166)(4,182)(5,183)(6,181)(7,169)(8,170)(9,171)(10,57)(11,55)(12,56)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,66)(20,64)(21,65)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(118,172)(119,173)(120,174), (1,86)(2,87)(3,85)(4,101)(5,102)(6,100)(7,88)(8,89)(9,90)(10,138)(11,136)(12,137)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,147)(20,145)(21,146)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,118)(38,119)(39,120)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(64,91)(65,92)(66,93)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(139,166)(140,167)(141,168)(142,169)(143,170)(144,171)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(172,199)(173,200)(174,201), (1,32)(2,33)(3,31)(4,47)(5,48)(6,46)(7,34)(8,35)(9,36)(10,192)(11,190)(12,191)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,201)(20,199)(21,200)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(37,64)(38,65)(39,66)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(91,118)(92,119)(93,120)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(145,172)(146,173)(147,174)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54)(55,56,57)(58,59,60)(61,62,63)(64,65,66)(67,68,69)(70,71,72)(73,74,75)(76,77,78)(79,80,81)(82,83,84)(85,86,87)(88,89,90)(91,92,93)(94,95,96)(97,98,99)(100,101,102)(103,104,105)(106,107,108)(109,110,111)(112,113,114)(115,116,117)(118,119,120)(121,122,123)(124,125,126)(127,128,129)(130,131,132)(133,134,135)(136,137,138)(139,140,141)(142,143,144)(145,146,147)(148,149,150)(151,152,153)(154,155,156)(157,158,159)(160,161,162)(163,164,165)(166,167,168)(169,170,171)(172,173,174)(175,176,177)(178,179,180)(181,182,183)(184,185,186)(187,188,189)(190,191,192)(193,194,195)(196,197,198)(199,200,201)(202,203,204)(205,206,207)(208,209,210)(211,212,213)(214,215,216), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,216,19)(11,214,20)(12,215,21)(25,37,28)(26,38,29)(27,39,30)(31,48,40)(32,46,41)(33,47,42)(34,51,43)(35,49,44)(36,50,45)(52,64,55)(53,65,56)(54,66,57)(58,75,67)(59,73,68)(60,74,69)(61,78,70)(62,76,71)(63,77,72)(79,91,82)(80,92,83)(81,93,84)(85,102,94)(86,100,95)(87,101,96)(88,105,97)(89,103,98)(90,104,99)(106,118,109)(107,119,110)(108,120,111)(112,129,121)(113,127,122)(114,128,123)(115,132,124)(116,130,125)(117,131,126)(133,145,136)(134,146,137)(135,147,138)(139,156,148)(140,154,149)(141,155,150)(142,159,151)(143,157,152)(144,158,153)(160,172,163)(161,173,164)(162,174,165)(166,183,175)(167,181,176)(168,182,177)(169,186,178)(170,184,179)(171,185,180)(187,199,190)(188,200,191)(189,201,192)(193,210,202)(194,208,203)(195,209,204)(196,213,205)(197,211,206)(198,212,207), (1,26,8)(2,27,9)(3,25,7)(4,39,23)(5,37,24)(6,38,22)(10,207,204)(11,205,202)(12,206,203)(13,28,16)(14,29,17)(15,30,18)(19,212,209)(20,213,210)(21,211,208)(31,52,34)(32,53,35)(33,54,36)(40,55,43)(41,56,44)(42,57,45)(46,65,49)(47,66,50)(48,64,51)(58,79,61)(59,80,62)(60,81,63)(67,82,70)(68,83,71)(69,84,72)(73,92,76)(74,93,77)(75,91,78)(85,106,88)(86,107,89)(87,108,90)(94,109,97)(95,110,98)(96,111,99)(100,119,103)(101,120,104)(102,118,105)(112,133,115)(113,134,116)(114,135,117)(121,136,124)(122,137,125)(123,138,126)(127,146,130)(128,147,131)(129,145,132)(139,160,142)(140,161,143)(141,162,144)(148,163,151)(149,164,152)(150,165,153)(154,173,157)(155,174,158)(156,172,159)(166,187,169)(167,188,170)(168,189,171)(175,190,178)(176,191,179)(177,192,180)(181,200,184)(182,201,185)(183,199,186)(193,214,196)(194,215,197)(195,216,198), (1,86)(2,85)(3,87)(4,94)(5,96)(6,95)(7,108)(8,107)(9,106)(10,132)(11,131)(12,130)(13,101)(14,100)(15,102)(16,120)(17,119)(18,118)(19,124)(20,126)(21,125)(22,110)(23,109)(24,111)(25,90)(26,89)(27,88)(28,104)(29,103)(30,105)(31,60)(32,59)(33,58)(34,81)(35,80)(36,79)(37,99)(38,98)(39,97)(40,74)(41,73)(42,75)(43,93)(44,92)(45,91)(46,68)(47,67)(48,69)(49,83)(50,82)(51,84)(52,63)(53,62)(54,61)(55,77)(56,76)(57,78)(64,72)(65,71)(66,70)(112,195)(113,194)(114,193)(115,216)(116,215)(117,214)(121,209)(122,208)(123,210)(127,203)(128,202)(129,204)(133,198)(134,197)(135,196)(136,212)(137,211)(138,213)(139,168)(140,167)(141,166)(142,189)(143,188)(144,187)(145,207)(146,206)(147,205)(148,182)(149,181)(150,183)(151,201)(152,200)(153,199)(154,176)(155,175)(156,177)(157,191)(158,190)(159,192)(160,171)(161,170)(162,169)(163,185)(164,184)(165,186)(172,180)(173,179)(174,178)>;
 
G:=Group( (1,167)(2,168)(3,166)(4,182)(5,183)(6,181)(7,169)(8,170)(9,171)(10,57)(11,55)(12,56)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,66)(20,64)(21,65)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(118,172)(119,173)(120,174), (1,86)(2,87)(3,85)(4,101)(5,102)(6,100)(7,88)(8,89)(9,90)(10,138)(11,136)(12,137)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,147)(20,145)(21,146)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,118)(38,119)(39,120)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(64,91)(65,92)(66,93)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(139,166)(140,167)(141,168)(142,169)(143,170)(144,171)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(172,199)(173,200)(174,201), (1,32)(2,33)(3,31)(4,47)(5,48)(6,46)(7,34)(8,35)(9,36)(10,192)(11,190)(12,191)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,201)(20,199)(21,200)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(37,64)(38,65)(39,66)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(91,118)(92,119)(93,120)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(145,172)(146,173)(147,174)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54)(55,56,57)(58,59,60)(61,62,63)(64,65,66)(67,68,69)(70,71,72)(73,74,75)(76,77,78)(79,80,81)(82,83,84)(85,86,87)(88,89,90)(91,92,93)(94,95,96)(97,98,99)(100,101,102)(103,104,105)(106,107,108)(109,110,111)(112,113,114)(115,116,117)(118,119,120)(121,122,123)(124,125,126)(127,128,129)(130,131,132)(133,134,135)(136,137,138)(139,140,141)(142,143,144)(145,146,147)(148,149,150)(151,152,153)(154,155,156)(157,158,159)(160,161,162)(163,164,165)(166,167,168)(169,170,171)(172,173,174)(175,176,177)(178,179,180)(181,182,183)(184,185,186)(187,188,189)(190,191,192)(193,194,195)(196,197,198)(199,200,201)(202,203,204)(205,206,207)(208,209,210)(211,212,213)(214,215,216), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,216,19)(11,214,20)(12,215,21)(25,37,28)(26,38,29)(27,39,30)(31,48,40)(32,46,41)(33,47,42)(34,51,43)(35,49,44)(36,50,45)(52,64,55)(53,65,56)(54,66,57)(58,75,67)(59,73,68)(60,74,69)(61,78,70)(62,76,71)(63,77,72)(79,91,82)(80,92,83)(81,93,84)(85,102,94)(86,100,95)(87,101,96)(88,105,97)(89,103,98)(90,104,99)(106,118,109)(107,119,110)(108,120,111)(112,129,121)(113,127,122)(114,128,123)(115,132,124)(116,130,125)(117,131,126)(133,145,136)(134,146,137)(135,147,138)(139,156,148)(140,154,149)(141,155,150)(142,159,151)(143,157,152)(144,158,153)(160,172,163)(161,173,164)(162,174,165)(166,183,175)(167,181,176)(168,182,177)(169,186,178)(170,184,179)(171,185,180)(187,199,190)(188,200,191)(189,201,192)(193,210,202)(194,208,203)(195,209,204)(196,213,205)(197,211,206)(198,212,207), (1,26,8)(2,27,9)(3,25,7)(4,39,23)(5,37,24)(6,38,22)(10,207,204)(11,205,202)(12,206,203)(13,28,16)(14,29,17)(15,30,18)(19,212,209)(20,213,210)(21,211,208)(31,52,34)(32,53,35)(33,54,36)(40,55,43)(41,56,44)(42,57,45)(46,65,49)(47,66,50)(48,64,51)(58,79,61)(59,80,62)(60,81,63)(67,82,70)(68,83,71)(69,84,72)(73,92,76)(74,93,77)(75,91,78)(85,106,88)(86,107,89)(87,108,90)(94,109,97)(95,110,98)(96,111,99)(100,119,103)(101,120,104)(102,118,105)(112,133,115)(113,134,116)(114,135,117)(121,136,124)(122,137,125)(123,138,126)(127,146,130)(128,147,131)(129,145,132)(139,160,142)(140,161,143)(141,162,144)(148,163,151)(149,164,152)(150,165,153)(154,173,157)(155,174,158)(156,172,159)(166,187,169)(167,188,170)(168,189,171)(175,190,178)(176,191,179)(177,192,180)(181,200,184)(182,201,185)(183,199,186)(193,214,196)(194,215,197)(195,216,198), (1,86)(2,85)(3,87)(4,94)(5,96)(6,95)(7,108)(8,107)(9,106)(10,132)(11,131)(12,130)(13,101)(14,100)(15,102)(16,120)(17,119)(18,118)(19,124)(20,126)(21,125)(22,110)(23,109)(24,111)(25,90)(26,89)(27,88)(28,104)(29,103)(30,105)(31,60)(32,59)(33,58)(34,81)(35,80)(36,79)(37,99)(38,98)(39,97)(40,74)(41,73)(42,75)(43,93)(44,92)(45,91)(46,68)(47,67)(48,69)(49,83)(50,82)(51,84)(52,63)(53,62)(54,61)(55,77)(56,76)(57,78)(64,72)(65,71)(66,70)(112,195)(113,194)(114,193)(115,216)(116,215)(117,214)(121,209)(122,208)(123,210)(127,203)(128,202)(129,204)(133,198)(134,197)(135,196)(136,212)(137,211)(138,213)(139,168)(140,167)(141,166)(142,189)(143,188)(144,187)(145,207)(146,206)(147,205)(148,182)(149,181)(150,183)(151,201)(152,200)(153,199)(154,176)(155,175)(156,177)(157,191)(158,190)(159,192)(160,171)(161,170)(162,169)(163,185)(164,184)(165,186)(172,180)(173,179)(174,178) );
 
G=PermutationGroup([[(1,167),(2,168),(3,166),(4,182),(5,183),(6,181),(7,169),(8,170),(9,171),(10,57),(11,55),(12,56),(13,175),(14,176),(15,177),(16,178),(17,179),(18,180),(19,66),(20,64),(21,65),(22,184),(23,185),(24,186),(25,187),(26,188),(27,189),(28,190),(29,191),(30,192),(31,193),(32,194),(33,195),(34,196),(35,197),(36,198),(37,199),(38,200),(39,201),(40,202),(41,203),(42,204),(43,205),(44,206),(45,207),(46,208),(47,209),(48,210),(49,211),(50,212),(51,213),(52,214),(53,215),(54,216),(58,112),(59,113),(60,114),(61,115),(62,116),(63,117),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134),(81,135),(82,136),(83,137),(84,138),(85,139),(86,140),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(118,172),(119,173),(120,174)], [(1,86),(2,87),(3,85),(4,101),(5,102),(6,100),(7,88),(8,89),(9,90),(10,138),(11,136),(12,137),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,147),(20,145),(21,146),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,118),(38,119),(39,120),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(57,84),(64,91),(65,92),(66,93),(112,193),(113,194),(114,195),(115,196),(116,197),(117,198),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(139,166),(140,167),(141,168),(142,169),(143,170),(144,171),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,181),(155,182),(156,183),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(172,199),(173,200),(174,201)], [(1,32),(2,33),(3,31),(4,47),(5,48),(6,46),(7,34),(8,35),(9,36),(10,192),(11,190),(12,191),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,201),(20,199),(21,200),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(37,64),(38,65),(39,66),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111),(91,118),(92,119),(93,120),(112,139),(113,140),(114,141),(115,142),(116,143),(117,144),(121,148),(122,149),(123,150),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,161),(135,162),(136,163),(137,164),(138,165),(145,172),(146,173),(147,174),(166,193),(167,194),(168,195),(169,196),(170,197),(171,198),(175,202),(176,203),(177,204),(178,205),(179,206),(180,207),(181,208),(182,209),(183,210),(184,211),(185,212),(186,213),(187,214),(188,215),(189,216)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36),(37,38,39),(40,41,42),(43,44,45),(46,47,48),(49,50,51),(52,53,54),(55,56,57),(58,59,60),(61,62,63),(64,65,66),(67,68,69),(70,71,72),(73,74,75),(76,77,78),(79,80,81),(82,83,84),(85,86,87),(88,89,90),(91,92,93),(94,95,96),(97,98,99),(100,101,102),(103,104,105),(106,107,108),(109,110,111),(112,113,114),(115,116,117),(118,119,120),(121,122,123),(124,125,126),(127,128,129),(130,131,132),(133,134,135),(136,137,138),(139,140,141),(142,143,144),(145,146,147),(148,149,150),(151,152,153),(154,155,156),(157,158,159),(160,161,162),(163,164,165),(166,167,168),(169,170,171),(172,173,174),(175,176,177),(178,179,180),(181,182,183),(184,185,186),(187,188,189),(190,191,192),(193,194,195),(196,197,198),(199,200,201),(202,203,204),(205,206,207),(208,209,210),(211,212,213),(214,215,216)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,216,19),(11,214,20),(12,215,21),(25,37,28),(26,38,29),(27,39,30),(31,48,40),(32,46,41),(33,47,42),(34,51,43),(35,49,44),(36,50,45),(52,64,55),(53,65,56),(54,66,57),(58,75,67),(59,73,68),(60,74,69),(61,78,70),(62,76,71),(63,77,72),(79,91,82),(80,92,83),(81,93,84),(85,102,94),(86,100,95),(87,101,96),(88,105,97),(89,103,98),(90,104,99),(106,118,109),(107,119,110),(108,120,111),(112,129,121),(113,127,122),(114,128,123),(115,132,124),(116,130,125),(117,131,126),(133,145,136),(134,146,137),(135,147,138),(139,156,148),(140,154,149),(141,155,150),(142,159,151),(143,157,152),(144,158,153),(160,172,163),(161,173,164),(162,174,165),(166,183,175),(167,181,176),(168,182,177),(169,186,178),(170,184,179),(171,185,180),(187,199,190),(188,200,191),(189,201,192),(193,210,202),(194,208,203),(195,209,204),(196,213,205),(197,211,206),(198,212,207)], [(1,26,8),(2,27,9),(3,25,7),(4,39,23),(5,37,24),(6,38,22),(10,207,204),(11,205,202),(12,206,203),(13,28,16),(14,29,17),(15,30,18),(19,212,209),(20,213,210),(21,211,208),(31,52,34),(32,53,35),(33,54,36),(40,55,43),(41,56,44),(42,57,45),(46,65,49),(47,66,50),(48,64,51),(58,79,61),(59,80,62),(60,81,63),(67,82,70),(68,83,71),(69,84,72),(73,92,76),(74,93,77),(75,91,78),(85,106,88),(86,107,89),(87,108,90),(94,109,97),(95,110,98),(96,111,99),(100,119,103),(101,120,104),(102,118,105),(112,133,115),(113,134,116),(114,135,117),(121,136,124),(122,137,125),(123,138,126),(127,146,130),(128,147,131),(129,145,132),(139,160,142),(140,161,143),(141,162,144),(148,163,151),(149,164,152),(150,165,153),(154,173,157),(155,174,158),(156,172,159),(166,187,169),(167,188,170),(168,189,171),(175,190,178),(176,191,179),(177,192,180),(181,200,184),(182,201,185),(183,199,186),(193,214,196),(194,215,197),(195,216,198)], [(1,86),(2,85),(3,87),(4,94),(5,96),(6,95),(7,108),(8,107),(9,106),(10,132),(11,131),(12,130),(13,101),(14,100),(15,102),(16,120),(17,119),(18,118),(19,124),(20,126),(21,125),(22,110),(23,109),(24,111),(25,90),(26,89),(27,88),(28,104),(29,103),(30,105),(31,60),(32,59),(33,58),(34,81),(35,80),(36,79),(37,99),(38,98),(39,97),(40,74),(41,73),(42,75),(43,93),(44,92),(45,91),(46,68),(47,67),(48,69),(49,83),(50,82),(51,84),(52,63),(53,62),(54,61),(55,77),(56,76),(57,78),(64,72),(65,71),(66,70),(112,195),(113,194),(114,193),(115,216),(116,215),(117,214),(121,209),(122,208),(123,210),(127,203),(128,202),(129,204),(133,198),(134,197),(135,196),(136,212),(137,211),(138,213),(139,168),(140,167),(141,166),(142,189),(143,188),(144,187),(145,207),(146,206),(147,205),(148,182),(149,181),(150,183),(151,201),(152,200),(153,199),(154,176),(155,175),(156,177),(157,191),(158,190),(159,192),(160,171),(161,170),(162,169),(163,185),(164,184),(165,186),(172,180),(173,179),(174,178)]])
 

120 conjugacy classes

class 1 2A···2G2H···2O3A···3M6A···6CM
order12···22···23···36···6
size11···127···272···22···2

120 irreducible representations

dim11122
type+++++
imageC1C2C2S3D6
kernelC23×C33⋊C2C22×C33⋊C2C63C2×C62C62
# reps11411391

Matrix representation of C23×C33⋊C2 ►in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
100000
010000
001000
000100
0000-10
00000-1
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
-1-10000
100000
000100
00-1-100
0000-1-1
000010
,
-1-10000
100000
001000
000100
000010
000001
,
010000
-1-10000
000100
00-1-100
000001
0000-1-1
,
0-10000
-100000
000-100
00-1000
000010
0000-1-1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1] >;
 

C23×C33⋊C2 in GAP, Magma, Sage, TeX 

C_2^3\times C_3^3\rtimes C_2
 
% in TeX
 
G:=Group("C2^3xC3^3:C2");
 
// GroupNames label
 
G:=SmallGroup(432,774);
 
// by ID
 
G=gap.SmallGroup(432,774);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,1124,4037,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^3=e^3=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,g*d*g=d^-1,e*f=f*e,g*e*g=e^-1,g*f*g=f^-1>;
 
// generators/relations
 

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