metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.29D10, C60.58C23, C30.34C24, C15⋊72- (1+4), Dic10.31D6, D30.40C23, Dic30.21C22, Dic15.20C23, (Q8×D5)⋊8S3, (Q8×D15)⋊5C2, Q8⋊3S3⋊4D5, (C4×D5).18D6, (C5×Q8).43D6, Q8.28(S3×D5), D12⋊D5⋊6C2, D12⋊5D5⋊6C2, C15⋊Q8.7C22, (S3×Dic10)⋊6C2, (C4×S3).18D10, (C3×Q8).26D10, C6.34(C23×D5), D6.D10⋊7C2, C10.34(S3×C23), C20.58(C22×S3), C5⋊3(Q8.15D6), D6.15(C22×D5), (C6×D5).48C23, C12.58(C22×D5), C15⋊D4.4C22, C5⋊D12.4C22, C3⋊D20.5C22, C3⋊4(D4.10D10), (S3×C20).21C22, (S3×C10).17C23, D10.44(C22×S3), (C5×D12).19C22, (D5×C12).21C22, (C4×D15).21C22, (Q8×C15).21C22, (S3×Dic5).4C22, (C5×Dic3).31C23, Dic3.28(C22×D5), Dic5.19(C22×S3), (C3×Dic5).17C23, (C3×Dic10).21C22, (C3×Q8×D5)⋊5C2, C4.58(C2×S3×D5), C2.37(C22×S3×D5), (C5×Q8⋊3S3)⋊4C2, SmallGroup(480,1106)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1324 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C4 [×7], C22 [×5], C5, S3 [×4], C6, C6, C2×C4 [×15], D4 [×10], Q8, Q8 [×9], D5 [×2], C10, C10 [×3], Dic3, Dic3 [×3], C12 [×3], C12 [×3], D6 [×3], D6, C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×3], Dic5 [×3], C20 [×3], C20, D10, D10, C2×C10 [×3], Dic6 [×6], C4×S3 [×3], C4×S3 [×9], D12 [×3], D12 [×3], C3⋊D4 [×4], C2×C12 [×3], C3×Q8, C3×Q8 [×3], C5×S3 [×3], C3×D5, D15, C30, 2- (1+4), Dic10 [×3], Dic10 [×6], C4×D5 [×3], C4×D5 [×3], D20, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C4○D12 [×6], S3×Q8 [×4], Q8⋊3S3, Q8⋊3S3 [×3], C6×Q8, C5×Dic3, C3×Dic5 [×3], Dic15 [×3], C60 [×3], C6×D5, S3×C10 [×3], D30, C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5 [×6], Q8×D5, Q8×D5, C5×C4○D4, Q8.15D6, S3×Dic5 [×6], C15⋊D4 [×3], C3⋊D20, C5⋊D12 [×3], C15⋊Q8 [×3], C3×Dic10 [×3], D5×C12 [×3], S3×C20 [×3], C5×D12 [×3], Dic30 [×3], C4×D15 [×3], Q8×C15, D4.10D10, S3×Dic10 [×3], D12⋊D5 [×3], D6.D10 [×3], D12⋊5D5 [×3], C3×Q8×D5, C5×Q8⋊3S3, Q8×D15, D12.29D10
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, D12.29D10
Generators and relations
G = < a,b,c,d | a12=b2=c10=1, d2=a6, bab=a-1, cac-1=dad-1=a5, cbc-1=a10b, dbd-1=a4b, dcd-1=a6c-1 >
►On 240 points
Generators in S240
(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 26)(2 25)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 120)(69 119)(70 118)(71 117)(72 116)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 108)(83 107)(84 106)(121 227)(122 226)(123 225)(124 224)(125 223)(126 222)(127 221)(128 220)(129 219)(130 218)(131 217)(132 228)(133 214)(134 213)(135 212)(136 211)(137 210)(138 209)(139 208)(140 207)(141 206)(142 205)(143 216)(144 215)(145 237)(146 236)(147 235)(148 234)(149 233)(150 232)(151 231)(152 230)(153 229)(154 240)(155 239)(156 238)(157 188)(158 187)(159 186)(160 185)(161 184)(162 183)(163 182)(164 181)(165 192)(166 191)(167 190)(168 189)(169 195)(170 194)(171 193)(172 204)(173 203)(174 202)(175 201)(176 200)(177 199)(178 198)(179 197)(180 196)
(1 176 99 66 188 210 51 239 130 23)(2 169 100 71 189 215 52 232 131 16)(3 174 101 64 190 208 53 237 132 21)(4 179 102 69 191 213 54 230 121 14)(5 172 103 62 192 206 55 235 122 19)(6 177 104 67 181 211 56 240 123 24)(7 170 105 72 182 216 57 233 124 17)(8 175 106 65 183 209 58 238 125 22)(9 180 107 70 184 214 59 231 126 15)(10 173 108 63 185 207 60 236 127 20)(11 178 97 68 186 212 49 229 128 13)(12 171 98 61 187 205 50 234 129 18)(25 197 78 119 168 134 87 152 217 41)(26 202 79 112 157 139 88 145 218 46)(27 195 80 117 158 144 89 150 219 39)(28 200 81 110 159 137 90 155 220 44)(29 193 82 115 160 142 91 148 221 37)(30 198 83 120 161 135 92 153 222 42)(31 203 84 113 162 140 93 146 223 47)(32 196 73 118 163 133 94 151 224 40)(33 201 74 111 164 138 95 156 225 45)(34 194 75 116 165 143 96 149 226 38)(35 199 76 109 166 136 85 154 227 43)(36 204 77 114 167 141 86 147 228 48)
(1 191 7 185)(2 184 8 190)(3 189 9 183)(4 182 10 188)(5 187 11 181)(6 192 12 186)(13 205 19 211)(14 210 20 216)(15 215 21 209)(16 208 22 214)(17 213 23 207)(18 206 24 212)(25 157 31 163)(26 162 32 168)(27 167 33 161)(28 160 34 166)(29 165 35 159)(30 158 36 164)(37 137 43 143)(38 142 44 136)(39 135 45 141)(40 140 46 134)(41 133 47 139)(42 138 48 144)(49 123 55 129)(50 128 56 122)(51 121 57 127)(52 126 58 132)(53 131 59 125)(54 124 60 130)(61 172 67 178)(62 177 68 171)(63 170 69 176)(64 175 70 169)(65 180 71 174)(66 173 72 179)(73 78 79 84)(74 83 80 77)(75 76 81 82)(85 220 91 226)(86 225 92 219)(87 218 93 224)(88 223 94 217)(89 228 95 222)(90 221 96 227)(97 104 103 98)(99 102 105 108)(100 107 106 101)(109 194 115 200)(110 199 116 193)(111 204 117 198)(112 197 118 203)(113 202 119 196)(114 195 120 201)(145 152 151 146)(147 150 153 156)(148 155 154 149)(229 234 235 240)(230 239 236 233)(231 232 237 238)
G:=sub<Sym(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), (1,26)(2,25)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,120)(69,119)(70,118)(71,117)(72,116)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,108)(83,107)(84,106)(121,227)(122,226)(123,225)(124,224)(125,223)(126,222)(127,221)(128,220)(129,219)(130,218)(131,217)(132,228)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,216)(144,215)(145,237)(146,236)(147,235)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)(154,240)(155,239)(156,238)(157,188)(158,187)(159,186)(160,185)(161,184)(162,183)(163,182)(164,181)(165,192)(166,191)(167,190)(168,189)(169,195)(170,194)(171,193)(172,204)(173,203)(174,202)(175,201)(176,200)(177,199)(178,198)(179,197)(180,196), (1,176,99,66,188,210,51,239,130,23)(2,169,100,71,189,215,52,232,131,16)(3,174,101,64,190,208,53,237,132,21)(4,179,102,69,191,213,54,230,121,14)(5,172,103,62,192,206,55,235,122,19)(6,177,104,67,181,211,56,240,123,24)(7,170,105,72,182,216,57,233,124,17)(8,175,106,65,183,209,58,238,125,22)(9,180,107,70,184,214,59,231,126,15)(10,173,108,63,185,207,60,236,127,20)(11,178,97,68,186,212,49,229,128,13)(12,171,98,61,187,205,50,234,129,18)(25,197,78,119,168,134,87,152,217,41)(26,202,79,112,157,139,88,145,218,46)(27,195,80,117,158,144,89,150,219,39)(28,200,81,110,159,137,90,155,220,44)(29,193,82,115,160,142,91,148,221,37)(30,198,83,120,161,135,92,153,222,42)(31,203,84,113,162,140,93,146,223,47)(32,196,73,118,163,133,94,151,224,40)(33,201,74,111,164,138,95,156,225,45)(34,194,75,116,165,143,96,149,226,38)(35,199,76,109,166,136,85,154,227,43)(36,204,77,114,167,141,86,147,228,48), (1,191,7,185)(2,184,8,190)(3,189,9,183)(4,182,10,188)(5,187,11,181)(6,192,12,186)(13,205,19,211)(14,210,20,216)(15,215,21,209)(16,208,22,214)(17,213,23,207)(18,206,24,212)(25,157,31,163)(26,162,32,168)(27,167,33,161)(28,160,34,166)(29,165,35,159)(30,158,36,164)(37,137,43,143)(38,142,44,136)(39,135,45,141)(40,140,46,134)(41,133,47,139)(42,138,48,144)(49,123,55,129)(50,128,56,122)(51,121,57,127)(52,126,58,132)(53,131,59,125)(54,124,60,130)(61,172,67,178)(62,177,68,171)(63,170,69,176)(64,175,70,169)(65,180,71,174)(66,173,72,179)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,220,91,226)(86,225,92,219)(87,218,93,224)(88,223,94,217)(89,228,95,222)(90,221,96,227)(97,104,103,98)(99,102,105,108)(100,107,106,101)(109,194,115,200)(110,199,116,193)(111,204,117,198)(112,197,118,203)(113,202,119,196)(114,195,120,201)(145,152,151,146)(147,150,153,156)(148,155,154,149)(229,234,235,240)(230,239,236,233)(231,232,237,238)>;
G:=Group( (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), (1,26)(2,25)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,120)(69,119)(70,118)(71,117)(72,116)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,108)(83,107)(84,106)(121,227)(122,226)(123,225)(124,224)(125,223)(126,222)(127,221)(128,220)(129,219)(130,218)(131,217)(132,228)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,216)(144,215)(145,237)(146,236)(147,235)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)(154,240)(155,239)(156,238)(157,188)(158,187)(159,186)(160,185)(161,184)(162,183)(163,182)(164,181)(165,192)(166,191)(167,190)(168,189)(169,195)(170,194)(171,193)(172,204)(173,203)(174,202)(175,201)(176,200)(177,199)(178,198)(179,197)(180,196), (1,176,99,66,188,210,51,239,130,23)(2,169,100,71,189,215,52,232,131,16)(3,174,101,64,190,208,53,237,132,21)(4,179,102,69,191,213,54,230,121,14)(5,172,103,62,192,206,55,235,122,19)(6,177,104,67,181,211,56,240,123,24)(7,170,105,72,182,216,57,233,124,17)(8,175,106,65,183,209,58,238,125,22)(9,180,107,70,184,214,59,231,126,15)(10,173,108,63,185,207,60,236,127,20)(11,178,97,68,186,212,49,229,128,13)(12,171,98,61,187,205,50,234,129,18)(25,197,78,119,168,134,87,152,217,41)(26,202,79,112,157,139,88,145,218,46)(27,195,80,117,158,144,89,150,219,39)(28,200,81,110,159,137,90,155,220,44)(29,193,82,115,160,142,91,148,221,37)(30,198,83,120,161,135,92,153,222,42)(31,203,84,113,162,140,93,146,223,47)(32,196,73,118,163,133,94,151,224,40)(33,201,74,111,164,138,95,156,225,45)(34,194,75,116,165,143,96,149,226,38)(35,199,76,109,166,136,85,154,227,43)(36,204,77,114,167,141,86,147,228,48), (1,191,7,185)(2,184,8,190)(3,189,9,183)(4,182,10,188)(5,187,11,181)(6,192,12,186)(13,205,19,211)(14,210,20,216)(15,215,21,209)(16,208,22,214)(17,213,23,207)(18,206,24,212)(25,157,31,163)(26,162,32,168)(27,167,33,161)(28,160,34,166)(29,165,35,159)(30,158,36,164)(37,137,43,143)(38,142,44,136)(39,135,45,141)(40,140,46,134)(41,133,47,139)(42,138,48,144)(49,123,55,129)(50,128,56,122)(51,121,57,127)(52,126,58,132)(53,131,59,125)(54,124,60,130)(61,172,67,178)(62,177,68,171)(63,170,69,176)(64,175,70,169)(65,180,71,174)(66,173,72,179)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,220,91,226)(86,225,92,219)(87,218,93,224)(88,223,94,217)(89,228,95,222)(90,221,96,227)(97,104,103,98)(99,102,105,108)(100,107,106,101)(109,194,115,200)(110,199,116,193)(111,204,117,198)(112,197,118,203)(113,202,119,196)(114,195,120,201)(145,152,151,146)(147,150,153,156)(148,155,154,149)(229,234,235,240)(230,239,236,233)(231,232,237,238) );
G=PermutationGroup([(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)], [(1,26),(2,25),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,115),(62,114),(63,113),(64,112),(65,111),(66,110),(67,109),(68,120),(69,119),(70,118),(71,117),(72,116),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,108),(83,107),(84,106),(121,227),(122,226),(123,225),(124,224),(125,223),(126,222),(127,221),(128,220),(129,219),(130,218),(131,217),(132,228),(133,214),(134,213),(135,212),(136,211),(137,210),(138,209),(139,208),(140,207),(141,206),(142,205),(143,216),(144,215),(145,237),(146,236),(147,235),(148,234),(149,233),(150,232),(151,231),(152,230),(153,229),(154,240),(155,239),(156,238),(157,188),(158,187),(159,186),(160,185),(161,184),(162,183),(163,182),(164,181),(165,192),(166,191),(167,190),(168,189),(169,195),(170,194),(171,193),(172,204),(173,203),(174,202),(175,201),(176,200),(177,199),(178,198),(179,197),(180,196)], [(1,176,99,66,188,210,51,239,130,23),(2,169,100,71,189,215,52,232,131,16),(3,174,101,64,190,208,53,237,132,21),(4,179,102,69,191,213,54,230,121,14),(5,172,103,62,192,206,55,235,122,19),(6,177,104,67,181,211,56,240,123,24),(7,170,105,72,182,216,57,233,124,17),(8,175,106,65,183,209,58,238,125,22),(9,180,107,70,184,214,59,231,126,15),(10,173,108,63,185,207,60,236,127,20),(11,178,97,68,186,212,49,229,128,13),(12,171,98,61,187,205,50,234,129,18),(25,197,78,119,168,134,87,152,217,41),(26,202,79,112,157,139,88,145,218,46),(27,195,80,117,158,144,89,150,219,39),(28,200,81,110,159,137,90,155,220,44),(29,193,82,115,160,142,91,148,221,37),(30,198,83,120,161,135,92,153,222,42),(31,203,84,113,162,140,93,146,223,47),(32,196,73,118,163,133,94,151,224,40),(33,201,74,111,164,138,95,156,225,45),(34,194,75,116,165,143,96,149,226,38),(35,199,76,109,166,136,85,154,227,43),(36,204,77,114,167,141,86,147,228,48)], [(1,191,7,185),(2,184,8,190),(3,189,9,183),(4,182,10,188),(5,187,11,181),(6,192,12,186),(13,205,19,211),(14,210,20,216),(15,215,21,209),(16,208,22,214),(17,213,23,207),(18,206,24,212),(25,157,31,163),(26,162,32,168),(27,167,33,161),(28,160,34,166),(29,165,35,159),(30,158,36,164),(37,137,43,143),(38,142,44,136),(39,135,45,141),(40,140,46,134),(41,133,47,139),(42,138,48,144),(49,123,55,129),(50,128,56,122),(51,121,57,127),(52,126,58,132),(53,131,59,125),(54,124,60,130),(61,172,67,178),(62,177,68,171),(63,170,69,176),(64,175,70,169),(65,180,71,174),(66,173,72,179),(73,78,79,84),(74,83,80,77),(75,76,81,82),(85,220,91,226),(86,225,92,219),(87,218,93,224),(88,223,94,217),(89,228,95,222),(90,221,96,227),(97,104,103,98),(99,102,105,108),(100,107,106,101),(109,194,115,200),(110,199,116,193),(111,204,117,198),(112,197,118,203),(113,202,119,196),(114,195,120,201),(145,152,151,146),(147,150,153,156),(148,155,154,149),(229,234,235,240),(230,239,236,233),(231,232,237,238)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 60 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 30 | 0 | 5 |
| 0 | 0 | 0 | 0 | 15 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 31 |
| 13 | 16 | 1 | 45 | 0 | 0 | 0 | 0 |
| 12 | 1 | 49 | 13 | 0 | 0 | 0 | 0 |
| 49 | 29 | 48 | 45 | 0 | 0 | 0 | 0 |
| 37 | 12 | 49 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 7 | 14 | 21 |
| 0 | 0 | 0 | 0 | 50 | 36 | 28 | 47 |
| 0 | 0 | 0 | 0 | 24 | 36 | 36 | 54 |
| 0 | 0 | 0 | 0 | 48 | 37 | 11 | 25 |
| 33 | 10 | 5 | 28 | 0 | 0 | 0 | 0 |
| 38 | 56 | 21 | 45 | 0 | 0 | 0 | 0 |
| 33 | 18 | 28 | 51 | 0 | 0 | 0 | 0 |
| 44 | 50 | 23 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 60 | 41 | 10 |
| 0 | 0 | 0 | 0 | 19 | 44 | 54 | 48 |
| 0 | 0 | 0 | 0 | 1 | 30 | 59 | 1 |
| 0 | 0 | 0 | 0 | 40 | 22 | 42 | 17 |
| 44 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 45 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 43 | 17 | 43 | 0 | 0 | 0 | 0 |
| 16 | 44 | 16 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 22 | 32 | 24 |
| 0 | 0 | 0 | 0 | 40 | 52 | 27 | 29 |
| 0 | 0 | 0 | 0 | 35 | 11 | 52 | 39 |
| 0 | 0 | 0 | 0 | 20 | 26 | 21 | 9 |
G:=sub<GL(8,GF(61))| [60,0,1,0,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,30,0,15,0,0,0,0,0,0,30,0,15,0,0,0,0,5,0,31,0,0,0,0,0,0,5,0,31],[13,12,49,37,0,0,0,0,16,1,29,12,0,0,0,0,1,49,48,49,0,0,0,0,45,13,45,60,0,0,0,0,0,0,0,0,25,50,24,48,0,0,0,0,7,36,36,37,0,0,0,0,14,28,36,11,0,0,0,0,21,47,54,25],[33,38,33,44,0,0,0,0,10,56,18,50,0,0,0,0,5,21,28,23,0,0,0,0,28,45,51,5,0,0,0,0,0,0,0,0,2,19,1,40,0,0,0,0,60,44,30,22,0,0,0,0,41,54,59,42,0,0,0,0,10,48,1,17],[44,45,17,16,0,0,0,0,18,17,43,44,0,0,0,0,0,0,17,16,0,0,0,0,0,0,43,44,0,0,0,0,0,0,0,0,9,40,35,20,0,0,0,0,22,52,11,26,0,0,0,0,32,27,52,21,0,0,0,0,24,29,39,9] >;
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 6 | 10 | 30 | 2 | 2 | 2 | 2 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | 2- (1+4) | S3×D5 | Q8.15D6 | C2×S3×D5 | D4.10D10 | D12.29D10 |
| kernel | D12.29D10 | S3×Dic10 | D12⋊D5 | D6.D10 | D12⋊5D5 | C3×Q8×D5 | C5×Q8⋊3S3 | Q8×D15 | Q8×D5 | Q8⋊3S3 | Dic10 | C4×D5 | C5×Q8 | C4×S3 | D12 | C3×Q8 | C15 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 1 | 2 | 2 | 6 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{12}._{29}D_{10} % in TeX
G:=Group("D12.29D10"); // GroupNames label
G:=SmallGroup(480,1106);
// by ID
G=gap.SmallGroup(480,1106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,100,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^10*b,d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀