direct product, metabelian, supersoluble, monomial, A-group
Aliases: C23×C9⋊S3, C62.151D6, (C3×C9)⋊4C24, (C2×C6)⋊12D18, (C2×C18)⋊12D6, C9⋊2(S3×C23), C3⋊2(C23×D9), (C22×C6)⋊5D9, C6⋊2(C22×D9), C18⋊2(C22×S3), (C22×C18)⋊5S3, (C3×C18)⋊4C23, (C6×C18)⋊14C22, (C2×C62).30S3, C32.4(S3×C23), (C2×C6×C18)⋊7C2, C3.(C23×C3⋊S3), C6.38(C22×C3⋊S3), (C22×C6).16(C3⋊S3), (C3×C6).173(C22×S3), (C2×C6).45(C2×C3⋊S3), SmallGroup(432,560)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C23×C9⋊S3 |
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
►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