metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.28D30, (C22×C20)⋊9S3, (C22×C60)⋊6C2, (C2×C4).68D30, (C22×C4)⋊5D15, (C22×C12)⋊5D5, D30⋊3C4⋊2C2, C30.377(C2×D4), (C2×C30).144D4, (C2×C20).383D6, C30.4Q8⋊3C2, (C2×C12).384D10, C30.38D4⋊6C2, C30.178(C4○D4), C6.106(C4○D20), (C2×C60).466C22, (C2×C30).303C23, (C22×C6).120D10, (C22×C10).138D6, C5⋊5(C23.28D6), C22.9(C15⋊7D4), C10.106(C4○D12), C3⋊5(C23.23D10), C15⋊34(C22.D4), C2.18(D60⋊11C2), C22.55(C22×D15), (C22×C30).143C22, (C2×Dic15).14C22, (C22×D15).10C22, C2.6(C2×C15⋊7D4), (C2×C15⋊7D4).6C2, C6.101(C2×C5⋊D4), (C2×C6).76(C5⋊D4), C10.101(C2×C3⋊D4), (C2×C10).76(C3⋊D4), (C2×C6).299(C22×D5), (C2×C10).298(C22×S3), SmallGroup(480,894)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.28D30
G = < a,b,c,d,e | a2=b2=c2=1, d30=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd29 >
Subgroups: 852 in 156 conjugacy classes, 55 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D15, C30, C30, C30, C22.D4, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, Dic15, C60, D30, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.28D6, C2×Dic15, C2×Dic15, C15⋊7D4, C2×C60, C2×C60, C22×D15, C22×C30, C23.23D10, C30.4Q8, D30⋊3C4, C30.38D4, C2×C15⋊7D4, C22×C60, C23.28D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, D15, C22.D4, C5⋊D4, C22×D5, C4○D12, C2×C3⋊D4, D30, C4○D20, C2×C5⋊D4, C23.28D6, C15⋊7D4, C22×D15, C23.23D10, D60⋊11C2, C2×C15⋊7D4, C23.28D30
►On 240 points
Generators in S240
(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 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 197)(135 198)(136 199)(137 200)(138 201)(139 202)(140 203)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 225)(163 226)(164 227)(165 228)(166 229)(167 230)(168 231)(169 232)(170 233)(171 234)(172 235)(173 236)(174 237)(175 238)(176 239)(177 240)(178 181)(179 182)(180 183)
(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 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)
(1 150 68 183)(2 182 69 149)(3 148 70 181)(4 240 71 147)(5 146 72 239)(6 238 73 145)(7 144 74 237)(8 236 75 143)(9 142 76 235)(10 234 77 141)(11 140 78 233)(12 232 79 139)(13 138 80 231)(14 230 81 137)(15 136 82 229)(16 228 83 135)(17 134 84 227)(18 226 85 133)(19 132 86 225)(20 224 87 131)(21 130 88 223)(22 222 89 129)(23 128 90 221)(24 220 91 127)(25 126 92 219)(26 218 93 125)(27 124 94 217)(28 216 95 123)(29 122 96 215)(30 214 97 121)(31 180 98 213)(32 212 99 179)(33 178 100 211)(34 210 101 177)(35 176 102 209)(36 208 103 175)(37 174 104 207)(38 206 105 173)(39 172 106 205)(40 204 107 171)(41 170 108 203)(42 202 109 169)(43 168 110 201)(44 200 111 167)(45 166 112 199)(46 198 113 165)(47 164 114 197)(48 196 115 163)(49 162 116 195)(50 194 117 161)(51 160 118 193)(52 192 119 159)(53 158 120 191)(54 190 61 157)(55 156 62 189)(56 188 63 155)(57 154 64 187)(58 186 65 153)(59 152 66 185)(60 184 67 151)
G:=sub<Sym(240)| (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), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,225)(163,226)(164,227)(165,228)(166,229)(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,181)(179,182)(180,183), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(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), (1,150,68,183)(2,182,69,149)(3,148,70,181)(4,240,71,147)(5,146,72,239)(6,238,73,145)(7,144,74,237)(8,236,75,143)(9,142,76,235)(10,234,77,141)(11,140,78,233)(12,232,79,139)(13,138,80,231)(14,230,81,137)(15,136,82,229)(16,228,83,135)(17,134,84,227)(18,226,85,133)(19,132,86,225)(20,224,87,131)(21,130,88,223)(22,222,89,129)(23,128,90,221)(24,220,91,127)(25,126,92,219)(26,218,93,125)(27,124,94,217)(28,216,95,123)(29,122,96,215)(30,214,97,121)(31,180,98,213)(32,212,99,179)(33,178,100,211)(34,210,101,177)(35,176,102,209)(36,208,103,175)(37,174,104,207)(38,206,105,173)(39,172,106,205)(40,204,107,171)(41,170,108,203)(42,202,109,169)(43,168,110,201)(44,200,111,167)(45,166,112,199)(46,198,113,165)(47,164,114,197)(48,196,115,163)(49,162,116,195)(50,194,117,161)(51,160,118,193)(52,192,119,159)(53,158,120,191)(54,190,61,157)(55,156,62,189)(56,188,63,155)(57,154,64,187)(58,186,65,153)(59,152,66,185)(60,184,67,151)>;
G:=Group( (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), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,225)(163,226)(164,227)(165,228)(166,229)(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,181)(179,182)(180,183), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(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), (1,150,68,183)(2,182,69,149)(3,148,70,181)(4,240,71,147)(5,146,72,239)(6,238,73,145)(7,144,74,237)(8,236,75,143)(9,142,76,235)(10,234,77,141)(11,140,78,233)(12,232,79,139)(13,138,80,231)(14,230,81,137)(15,136,82,229)(16,228,83,135)(17,134,84,227)(18,226,85,133)(19,132,86,225)(20,224,87,131)(21,130,88,223)(22,222,89,129)(23,128,90,221)(24,220,91,127)(25,126,92,219)(26,218,93,125)(27,124,94,217)(28,216,95,123)(29,122,96,215)(30,214,97,121)(31,180,98,213)(32,212,99,179)(33,178,100,211)(34,210,101,177)(35,176,102,209)(36,208,103,175)(37,174,104,207)(38,206,105,173)(39,172,106,205)(40,204,107,171)(41,170,108,203)(42,202,109,169)(43,168,110,201)(44,200,111,167)(45,166,112,199)(46,198,113,165)(47,164,114,197)(48,196,115,163)(49,162,116,195)(50,194,117,161)(51,160,118,193)(52,192,119,159)(53,158,120,191)(54,190,61,157)(55,156,62,189)(56,188,63,155)(57,154,64,187)(58,186,65,153)(59,152,66,185)(60,184,67,151) );
G=PermutationGroup([[(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)], [(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,97),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,191),(129,192),(130,193),(131,194),(132,195),(133,196),(134,197),(135,198),(136,199),(137,200),(138,201),(139,202),(140,203),(141,204),(142,205),(143,206),(144,207),(145,208),(146,209),(147,210),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,217),(155,218),(156,219),(157,220),(158,221),(159,222),(160,223),(161,224),(162,225),(163,226),(164,227),(165,228),(166,229),(167,230),(168,231),(169,232),(170,233),(171,234),(172,235),(173,236),(174,237),(175,238),(176,239),(177,240),(178,181),(179,182),(180,183)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(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)], [(1,150,68,183),(2,182,69,149),(3,148,70,181),(4,240,71,147),(5,146,72,239),(6,238,73,145),(7,144,74,237),(8,236,75,143),(9,142,76,235),(10,234,77,141),(11,140,78,233),(12,232,79,139),(13,138,80,231),(14,230,81,137),(15,136,82,229),(16,228,83,135),(17,134,84,227),(18,226,85,133),(19,132,86,225),(20,224,87,131),(21,130,88,223),(22,222,89,129),(23,128,90,221),(24,220,91,127),(25,126,92,219),(26,218,93,125),(27,124,94,217),(28,216,95,123),(29,122,96,215),(30,214,97,121),(31,180,98,213),(32,212,99,179),(33,178,100,211),(34,210,101,177),(35,176,102,209),(36,208,103,175),(37,174,104,207),(38,206,105,173),(39,172,106,205),(40,204,107,171),(41,170,108,203),(42,202,109,169),(43,168,110,201),(44,200,111,167),(45,166,112,199),(46,198,113,165),(47,164,114,197),(48,196,115,163),(49,162,116,195),(50,194,117,161),(51,160,118,193),(52,192,119,159),(53,158,120,191),(54,190,61,157),(55,156,62,189),(56,188,63,155),(57,154,64,187),(58,186,65,153),(59,152,66,185),(60,184,67,151)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 60 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 60 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D15 | C5⋊D4 | C4○D12 | D30 | D30 | C4○D20 | C15⋊7D4 | D60⋊11C2 |
| kernel | C23.28D30 | C30.4Q8 | D30⋊3C4 | C30.38D4 | C2×C15⋊7D4 | C22×C60 | C22×C20 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C10 | C2×C4 | C23 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C23.28D30 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 35 | 0 | 0 |
| 0 | 0 | 5 | 32 |
| 0 | 0 | 29 | 39 |
| 0 | 26 | 0 | 0 |
| 54 | 0 | 0 | 0 |
| 0 | 0 | 50 | 46 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,35,0,0,0,0,5,29,0,0,32,39],[0,54,0,0,26,0,0,0,0,0,50,0,0,0,46,11] >;
C23.28D30 in GAP, Magma, Sage, TeX
C_2^3._{28}D_{30} % in TeX
G:=Group("C2^3.28D30"); // GroupNames label
G:=SmallGroup(480,894);
// by ID
G=gap.SmallGroup(480,894);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^29>;
// generators/relations