metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8D30, C60⋊5C4⋊9C2, (C2×C4).26D30, (C2×C12).32D10, (C2×C20).206D6, C22⋊C4.2D15, C30.4Q8⋊5C2, (C4×Dic15)⋊16C2, C6.95(C4○D20), (C2×C60).17C22, (C22×C6).54D10, (C22×C10).69D6, C5⋊7(C23.8D6), C15⋊18(C42⋊2C2), C30.215(C4○D4), C10.95(C4○D12), C6.90(D4⋊2D5), C2.7(D4⋊2D15), (C2×C30).276C23, C30.38D4.3C2, C3⋊7(C23.D10), C10.90(D4⋊2S3), C2.9(D60⋊11C2), (C22×C30).10C22, (C2×Dic15).6C22, C22.40(C22×D15), (C5×C22⋊C4).2S3, (C3×C22⋊C4).2D5, (C15×C22⋊C4).2C2, (C2×C6).272(C22×D5), (C2×C10).271(C22×S3), SmallGroup(480,844)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.8D30
G = < a,b,c,d,e | a2=b2=c2=1, d30=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >
Subgroups: 548 in 120 conjugacy classes, 47 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C30 [×3], C30, C42⋊2C2, C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, Dic15 [×4], C60 [×2], C2×C30, C2×C30 [×3], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C23.8D6, C2×Dic15 [×4], C2×C60 [×2], C22×C30, C23.D10, C4×Dic15, C30.4Q8 [×2], C60⋊5C4, C30.38D4 [×2], C15×C22⋊C4, C23.8D30
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, D15, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3 [×2], D30 [×3], C4○D20, D4⋊2D5 [×2], C23.8D6, C22×D15, C23.D10, D60⋊11C2, D4⋊2D15 [×2], C23.8D30
►On 240 points
Generators in S240
(2 173)(4 175)(6 177)(8 179)(10 121)(12 123)(14 125)(16 127)(18 129)(20 131)(22 133)(24 135)(26 137)(28 139)(30 141)(32 143)(34 145)(36 147)(38 149)(40 151)(42 153)(44 155)(46 157)(48 159)(50 161)(52 163)(54 165)(56 167)(58 169)(60 171)(61 91)(62 224)(63 93)(64 226)(65 95)(66 228)(67 97)(68 230)(69 99)(70 232)(71 101)(72 234)(73 103)(74 236)(75 105)(76 238)(77 107)(78 240)(79 109)(80 182)(81 111)(82 184)(83 113)(84 186)(85 115)(86 188)(87 117)(88 190)(89 119)(90 192)(92 194)(94 196)(96 198)(98 200)(100 202)(102 204)(104 206)(106 208)(108 210)(110 212)(112 214)(114 216)(116 218)(118 220)(120 222)(181 211)(183 213)(185 215)(187 217)(189 219)(191 221)(193 223)(195 225)(197 227)(199 229)(201 231)(203 233)(205 235)(207 237)(209 239)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 121)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 161)(51 162)(52 163)(53 164)(54 165)(55 166)(56 167)(57 168)(58 169)(59 170)(60 171)(61 193)(62 194)(63 195)(64 196)(65 197)(66 198)(67 199)(68 200)(69 201)(70 202)(71 203)(72 204)(73 205)(74 206)(75 207)(76 208)(77 209)(78 210)(79 211)(80 212)(81 213)(82 214)(83 215)(84 216)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 225)(94 226)(95 227)(96 228)(97 229)(98 230)(99 231)(100 232)(101 233)(102 234)(103 235)(104 236)(105 237)(106 238)(107 239)(108 240)(109 181)(110 182)(111 183)(112 184)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 115 142 217)(2 84 143 186)(3 113 144 215)(4 82 145 184)(5 111 146 213)(6 80 147 182)(7 109 148 211)(8 78 149 240)(9 107 150 209)(10 76 151 238)(11 105 152 207)(12 74 153 236)(13 103 154 205)(14 72 155 234)(15 101 156 203)(16 70 157 232)(17 99 158 201)(18 68 159 230)(19 97 160 199)(20 66 161 228)(21 95 162 197)(22 64 163 226)(23 93 164 195)(24 62 165 224)(25 91 166 193)(26 120 167 222)(27 89 168 191)(28 118 169 220)(29 87 170 189)(30 116 171 218)(31 85 172 187)(32 114 173 216)(33 83 174 185)(34 112 175 214)(35 81 176 183)(36 110 177 212)(37 79 178 181)(38 108 179 210)(39 77 180 239)(40 106 121 208)(41 75 122 237)(42 104 123 206)(43 73 124 235)(44 102 125 204)(45 71 126 233)(46 100 127 202)(47 69 128 231)(48 98 129 200)(49 67 130 229)(50 96 131 198)(51 65 132 227)(52 94 133 196)(53 63 134 225)(54 92 135 194)(55 61 136 223)(56 90 137 192)(57 119 138 221)(58 88 139 190)(59 117 140 219)(60 86 141 188)
G:=sub<Sym(240)| (2,173)(4,175)(6,177)(8,179)(10,121)(12,123)(14,125)(16,127)(18,129)(20,131)(22,133)(24,135)(26,137)(28,139)(30,141)(32,143)(34,145)(36,147)(38,149)(40,151)(42,153)(44,155)(46,157)(48,159)(50,161)(52,163)(54,165)(56,167)(58,169)(60,171)(61,91)(62,224)(63,93)(64,226)(65,95)(66,228)(67,97)(68,230)(69,99)(70,232)(71,101)(72,234)(73,103)(74,236)(75,105)(76,238)(77,107)(78,240)(79,109)(80,182)(81,111)(82,184)(83,113)(84,186)(85,115)(86,188)(87,117)(88,190)(89,119)(90,192)(92,194)(94,196)(96,198)(98,200)(100,202)(102,204)(104,206)(106,208)(108,210)(110,212)(112,214)(114,216)(116,218)(118,220)(120,222)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,225)(94,226)(95,227)(96,228)(97,229)(98,230)(99,231)(100,232)(101,233)(102,234)(103,235)(104,236)(105,237)(106,238)(107,239)(108,240)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,115,142,217)(2,84,143,186)(3,113,144,215)(4,82,145,184)(5,111,146,213)(6,80,147,182)(7,109,148,211)(8,78,149,240)(9,107,150,209)(10,76,151,238)(11,105,152,207)(12,74,153,236)(13,103,154,205)(14,72,155,234)(15,101,156,203)(16,70,157,232)(17,99,158,201)(18,68,159,230)(19,97,160,199)(20,66,161,228)(21,95,162,197)(22,64,163,226)(23,93,164,195)(24,62,165,224)(25,91,166,193)(26,120,167,222)(27,89,168,191)(28,118,169,220)(29,87,170,189)(30,116,171,218)(31,85,172,187)(32,114,173,216)(33,83,174,185)(34,112,175,214)(35,81,176,183)(36,110,177,212)(37,79,178,181)(38,108,179,210)(39,77,180,239)(40,106,121,208)(41,75,122,237)(42,104,123,206)(43,73,124,235)(44,102,125,204)(45,71,126,233)(46,100,127,202)(47,69,128,231)(48,98,129,200)(49,67,130,229)(50,96,131,198)(51,65,132,227)(52,94,133,196)(53,63,134,225)(54,92,135,194)(55,61,136,223)(56,90,137,192)(57,119,138,221)(58,88,139,190)(59,117,140,219)(60,86,141,188)>;
G:=Group( (2,173)(4,175)(6,177)(8,179)(10,121)(12,123)(14,125)(16,127)(18,129)(20,131)(22,133)(24,135)(26,137)(28,139)(30,141)(32,143)(34,145)(36,147)(38,149)(40,151)(42,153)(44,155)(46,157)(48,159)(50,161)(52,163)(54,165)(56,167)(58,169)(60,171)(61,91)(62,224)(63,93)(64,226)(65,95)(66,228)(67,97)(68,230)(69,99)(70,232)(71,101)(72,234)(73,103)(74,236)(75,105)(76,238)(77,107)(78,240)(79,109)(80,182)(81,111)(82,184)(83,113)(84,186)(85,115)(86,188)(87,117)(88,190)(89,119)(90,192)(92,194)(94,196)(96,198)(98,200)(100,202)(102,204)(104,206)(106,208)(108,210)(110,212)(112,214)(114,216)(116,218)(118,220)(120,222)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,225)(94,226)(95,227)(96,228)(97,229)(98,230)(99,231)(100,232)(101,233)(102,234)(103,235)(104,236)(105,237)(106,238)(107,239)(108,240)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,115,142,217)(2,84,143,186)(3,113,144,215)(4,82,145,184)(5,111,146,213)(6,80,147,182)(7,109,148,211)(8,78,149,240)(9,107,150,209)(10,76,151,238)(11,105,152,207)(12,74,153,236)(13,103,154,205)(14,72,155,234)(15,101,156,203)(16,70,157,232)(17,99,158,201)(18,68,159,230)(19,97,160,199)(20,66,161,228)(21,95,162,197)(22,64,163,226)(23,93,164,195)(24,62,165,224)(25,91,166,193)(26,120,167,222)(27,89,168,191)(28,118,169,220)(29,87,170,189)(30,116,171,218)(31,85,172,187)(32,114,173,216)(33,83,174,185)(34,112,175,214)(35,81,176,183)(36,110,177,212)(37,79,178,181)(38,108,179,210)(39,77,180,239)(40,106,121,208)(41,75,122,237)(42,104,123,206)(43,73,124,235)(44,102,125,204)(45,71,126,233)(46,100,127,202)(47,69,128,231)(48,98,129,200)(49,67,130,229)(50,96,131,198)(51,65,132,227)(52,94,133,196)(53,63,134,225)(54,92,135,194)(55,61,136,223)(56,90,137,192)(57,119,138,221)(58,88,139,190)(59,117,140,219)(60,86,141,188) );
G=PermutationGroup([(2,173),(4,175),(6,177),(8,179),(10,121),(12,123),(14,125),(16,127),(18,129),(20,131),(22,133),(24,135),(26,137),(28,139),(30,141),(32,143),(34,145),(36,147),(38,149),(40,151),(42,153),(44,155),(46,157),(48,159),(50,161),(52,163),(54,165),(56,167),(58,169),(60,171),(61,91),(62,224),(63,93),(64,226),(65,95),(66,228),(67,97),(68,230),(69,99),(70,232),(71,101),(72,234),(73,103),(74,236),(75,105),(76,238),(77,107),(78,240),(79,109),(80,182),(81,111),(82,184),(83,113),(84,186),(85,115),(86,188),(87,117),(88,190),(89,119),(90,192),(92,194),(94,196),(96,198),(98,200),(100,202),(102,204),(104,206),(106,208),(108,210),(110,212),(112,214),(114,216),(116,218),(118,220),(120,222),(181,211),(183,213),(185,215),(187,217),(189,219),(191,221),(193,223),(195,225),(197,227),(199,229),(201,231),(203,233),(205,235),(207,237),(209,239)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)], [(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,121),(11,122),(12,123),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,151),(41,152),(42,153),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,160),(50,161),(51,162),(52,163),(53,164),(54,165),(55,166),(56,167),(57,168),(58,169),(59,170),(60,171),(61,193),(62,194),(63,195),(64,196),(65,197),(66,198),(67,199),(68,200),(69,201),(70,202),(71,203),(72,204),(73,205),(74,206),(75,207),(76,208),(77,209),(78,210),(79,211),(80,212),(81,213),(82,214),(83,215),(84,216),(85,217),(86,218),(87,219),(88,220),(89,221),(90,222),(91,223),(92,224),(93,225),(94,226),(95,227),(96,228),(97,229),(98,230),(99,231),(100,232),(101,233),(102,234),(103,235),(104,236),(105,237),(106,238),(107,239),(108,240),(109,181),(110,182),(111,183),(112,184),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,115,142,217),(2,84,143,186),(3,113,144,215),(4,82,145,184),(5,111,146,213),(6,80,147,182),(7,109,148,211),(8,78,149,240),(9,107,150,209),(10,76,151,238),(11,105,152,207),(12,74,153,236),(13,103,154,205),(14,72,155,234),(15,101,156,203),(16,70,157,232),(17,99,158,201),(18,68,159,230),(19,97,160,199),(20,66,161,228),(21,95,162,197),(22,64,163,226),(23,93,164,195),(24,62,165,224),(25,91,166,193),(26,120,167,222),(27,89,168,191),(28,118,169,220),(29,87,170,189),(30,116,171,218),(31,85,172,187),(32,114,173,216),(33,83,174,185),(34,112,175,214),(35,81,176,183),(36,110,177,212),(37,79,178,181),(38,108,179,210),(39,77,180,239),(40,106,121,208),(41,75,122,237),(42,104,123,206),(43,73,124,235),(44,102,125,204),(45,71,126,233),(46,100,127,202),(47,69,128,231),(48,98,129,200),(49,67,130,229),(50,96,131,198),(51,65,132,227),(52,94,133,196),(53,63,134,225),(54,92,135,194),(55,61,136,223),(56,90,137,192),(57,119,138,221),(58,88,139,190),(59,117,140,219),(60,86,141,188)])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 30 | 30 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | C4○D12 | D30 | D30 | C4○D20 | D60⋊11C2 | D4⋊2S3 | D4⋊2D5 | D4⋊2D15 |
| kernel | C23.8D30 | C4×Dic15 | C30.4Q8 | C60⋊5C4 | C30.38D4 | C15×C22⋊C4 | C5×C22⋊C4 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C22⋊C4 | C10 | C2×C4 | C23 | C6 | C2 | C10 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 6 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 2 | 4 | 8 |
Matrix representation of C23.8D30 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 41 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 21 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 53 |
| 0 | 0 | 0 | 0 | 45 | 19 |
| 4 | 53 | 0 | 0 | 0 | 0 |
| 40 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 22 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,1,41,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[46,21,0,0,0,0,0,4,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,42,45,0,0,0,0,53,19],[4,40,0,0,0,0,53,57,0,0,0,0,0,0,50,0,0,0,0,0,22,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;
C23.8D30 in GAP, Magma, Sage, TeX
C_2^3._8D_{30} % in TeX
G:=Group("C2^3.8D30"); // GroupNames label
G:=SmallGroup(480,844);
// by ID
G=gap.SmallGroup(480,844);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^29>;
// generators/relations