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## G = C23×C9⋊S3order 432 = 24·33

### Direct product of C23 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C23×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊S3 — C2×C9⋊S3 — C22×C9⋊S3 — C23×C9⋊S3
 Lower central C3×C9 — C23×C9⋊S3
 Upper central C1 — C23

Generators and relations for C23×C9⋊S3
G = < a,b,c,d,e,f | a2=b2=c2=d9=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 4036 in 670 conjugacy classes, 211 normal (9 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C23, C23, C9, C32, D6, C2×C6, C24, D9, C18, C3⋊S3, C3×C6, C22×S3, C22×C6, C22×C6, C3×C9, D18, C2×C18, C2×C3⋊S3, C62, S3×C23, C9⋊S3, C3×C18, C22×D9, C22×C18, C22×C3⋊S3, C2×C62, C2×C9⋊S3, C6×C18, C23×D9, C23×C3⋊S3, C22×C9⋊S3, C2×C6×C18, C23×C9⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, S3×C23, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, C23×D9, C23×C3⋊S3, C22×C9⋊S3, C23×C9⋊S3

Smallest permutation representation of C23×C9⋊S3
On 216 points
Generators in S216
(1 211)(2 212)(3 213)(4 214)(5 215)(6 216)(7 208)(8 209)(9 210)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(46 181)(47 182)(48 183)(49 184)(50 185)(51 186)(52 187)(53 188)(54 189)(55 190)(56 191)(57 192)(58 193)(59 194)(60 195)(61 196)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 145)(119 146)(120 147)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 100)(8 101)(9 102)(10 140)(11 141)(12 142)(13 143)(14 144)(15 136)(16 137)(17 138)(18 139)(19 152)(20 153)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 193)(167 194)(168 195)(169 196)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 194)(11 195)(12 196)(13 197)(14 198)(15 190)(16 191)(17 192)(18 193)(19 206)(20 207)(21 199)(22 200)(23 201)(24 202)(25 203)(26 204)(27 205)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(141 168)(142 169)(143 170)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(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 32 37)(2 33 38)(3 34 39)(4 35 40)(5 36 41)(6 28 42)(7 29 43)(8 30 44)(9 31 45)(10 21 211)(11 22 212)(12 23 213)(13 24 214)(14 25 215)(15 26 216)(16 27 208)(17 19 209)(18 20 210)(46 56 70)(47 57 71)(48 58 72)(49 59 64)(50 60 65)(51 61 66)(52 62 67)(53 63 68)(54 55 69)(73 83 97)(74 84 98)(75 85 99)(76 86 91)(77 87 92)(78 88 93)(79 89 94)(80 90 95)(81 82 96)(100 110 124)(101 111 125)(102 112 126)(103 113 118)(104 114 119)(105 115 120)(106 116 121)(107 117 122)(108 109 123)(127 137 151)(128 138 152)(129 139 153)(130 140 145)(131 141 146)(132 142 147)(133 143 148)(134 144 149)(135 136 150)(154 164 178)(155 165 179)(156 166 180)(157 167 172)(158 168 173)(159 169 174)(160 170 175)(161 171 176)(162 163 177)(181 191 205)(182 192 206)(183 193 207)(184 194 199)(185 195 200)(186 196 201)(187 197 202)(188 198 203)(189 190 204)
(1 157)(2 156)(3 155)(4 154)(5 162)(6 161)(7 160)(8 159)(9 158)(10 91)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 90)(27 89)(28 176)(29 175)(30 174)(31 173)(32 172)(33 180)(34 179)(35 178)(36 177)(37 167)(38 166)(39 165)(40 164)(41 163)(42 171)(43 170)(44 169)(45 168)(46 133)(47 132)(48 131)(49 130)(50 129)(51 128)(52 127)(53 135)(54 134)(55 149)(56 148)(57 147)(58 146)(59 145)(60 153)(61 152)(62 151)(63 150)(64 140)(65 139)(66 138)(67 137)(68 136)(69 144)(70 143)(71 142)(72 141)(73 214)(74 213)(75 212)(76 211)(77 210)(78 209)(79 208)(80 216)(81 215)(100 187)(101 186)(102 185)(103 184)(104 183)(105 182)(106 181)(107 189)(108 188)(109 203)(110 202)(111 201)(112 200)(113 199)(114 207)(115 206)(116 205)(117 204)(118 194)(119 193)(120 192)(121 191)(122 190)(123 198)(124 197)(125 196)(126 195)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 6A ··· 6AB 9A ··· 9I 18A ··· 18BK order 1 2 ··· 2 2 ··· 2 3 3 3 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 27 ··· 27 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D18 kernel C23×C9⋊S3 C22×C9⋊S3 C2×C6×C18 C22×C18 C2×C62 C2×C18 C62 C22×C6 C2×C6 # reps 1 14 1 3 1 21 7 9 63

Matrix representation of C23×C9⋊S3 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 7 14 0 0 0 5 2 0 0 0 0 0 17 7 0 0 0 12 5
,
 1 0 0 0 0 0 0 18 0 0 0 1 18 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 18 0 0 0 18 0 0 0 0 0 0 1 18 0 0 0 0 18

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,7,5,0,0,0,14,2,0,0,0,0,0,17,12,0,0,0,7,5],[1,0,0,0,0,0,0,1,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,0,18,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,18,18] >;

C23×C9⋊S3 in GAP, Magma, Sage, TeX

C_2^3\times C_9\rtimes S_3
% in TeX

G:=Group("C2^3xC9:S3");
// GroupNames label

G:=SmallGroup(432,560);
// by ID

G=gap.SmallGroup(432,560);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,6164,662,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^9=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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