direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12×C5⋊D4, C60⋊33D4, C5⋊5(D4×C12), C20⋊8(C3×D4), C15⋊41(C4×D4), D10⋊5(C2×C12), C10.45(C6×D4), C22⋊3(D5×C12), (C22×C12)⋊2D5, (C22×C60)⋊21C2, (C22×C20)⋊13C6, (C4×Dic5)⋊16C6, Dic5⋊3(C2×C12), C30.399(C2×D4), C23.D5⋊14C6, D10⋊C4⋊18C6, C23.27(C6×D5), (C12×Dic5)⋊34C2, (C2×C12).454D10, C10.D4⋊18C6, C30.194(C4○D4), C6.124(C4○D20), C10.33(C22×C12), (C2×C30).363C23, C30.191(C22×C4), (C2×C60).553C22, (C22×C6).107D10, (C22×C30).159C22, (C6×Dic5).248C22, (C2×C4×D5)⋊14C6, (D5×C2×C12)⋊30C2, (C2×C6)⋊10(C4×D5), C2.20(D5×C2×C12), C6.116(C2×C4×D5), C2.3(C6×C5⋊D4), (C2×C30)⋊35(C2×C4), (C6×D5)⋊25(C2×C4), C2.5(C3×C4○D20), (C2×C4).65(C6×D5), (C2×C5⋊D4).9C6, (C22×C4)⋊4(C3×D5), (C2×C10)⋊12(C2×C12), C22.24(D5×C2×C6), (C2×C20).79(C2×C6), C10.15(C3×C4○D4), (C6×C5⋊D4).18C2, C6.126(C2×C5⋊D4), (C3×Dic5)⋊18(C2×C4), (C3×C23.D5)⋊30C2, (C3×D10⋊C4)⋊40C2, (D5×C2×C6).134C22, (C3×C10.D4)⋊40C2, (C22×C10).46(C2×C6), (C2×C10).46(C22×C6), (C2×Dic5).38(C2×C6), (C22×D5).29(C2×C6), (C2×C6).359(C22×D5), SmallGroup(480,721)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×C5⋊D4
G = < a,b,c,d | a12=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 528 in 188 conjugacy classes, 90 normal (58 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4×D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, D4×C12, D5×C12, C6×Dic5, C3×C5⋊D4, C2×C60, C2×C60, D5×C2×C6, C22×C30, C4×C5⋊D4, C12×Dic5, C3×C10.D4, C3×D10⋊C4, C3×C23.D5, D5×C2×C12, C6×C5⋊D4, C22×C60, C12×C5⋊D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22×C4, C2×D4, C4○D4, D10, C2×C12, C3×D4, C22×C6, C3×D5, C4×D4, C4×D5, C5⋊D4, C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, D4×C12, D5×C12, C3×C5⋊D4, D5×C2×C6, C4×C5⋊D4, D5×C2×C12, C3×C4○D20, C6×C5⋊D4, C12×C5⋊D4
►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)
(1 105 57 204 25)(2 106 58 193 26)(3 107 59 194 27)(4 108 60 195 28)(5 97 49 196 29)(6 98 50 197 30)(7 99 51 198 31)(8 100 52 199 32)(9 101 53 200 33)(10 102 54 201 34)(11 103 55 202 35)(12 104 56 203 36)(13 166 173 139 212)(14 167 174 140 213)(15 168 175 141 214)(16 157 176 142 215)(17 158 177 143 216)(18 159 178 144 205)(19 160 179 133 206)(20 161 180 134 207)(21 162 169 135 208)(22 163 170 136 209)(23 164 171 137 210)(24 165 172 138 211)(37 225 119 89 131)(38 226 120 90 132)(39 227 109 91 121)(40 228 110 92 122)(41 217 111 93 123)(42 218 112 94 124)(43 219 113 95 125)(44 220 114 96 126)(45 221 115 85 127)(46 222 116 86 128)(47 223 117 87 129)(48 224 118 88 130)(61 147 183 238 73)(62 148 184 239 74)(63 149 185 240 75)(64 150 186 229 76)(65 151 187 230 77)(66 152 188 231 78)(67 153 189 232 79)(68 154 190 233 80)(69 155 191 234 81)(70 156 192 235 82)(71 145 181 236 83)(72 146 182 237 84)
(1 158 147 224)(2 159 148 225)(3 160 149 226)(4 161 150 227)(5 162 151 228)(6 163 152 217)(7 164 153 218)(8 165 154 219)(9 166 155 220)(10 167 156 221)(11 168 145 222)(12 157 146 223)(13 191 44 101)(14 192 45 102)(15 181 46 103)(16 182 47 104)(17 183 48 105)(18 184 37 106)(19 185 38 107)(20 186 39 108)(21 187 40 97)(22 188 41 98)(23 189 42 99)(24 190 43 100)(25 177 61 118)(26 178 62 119)(27 179 63 120)(28 180 64 109)(29 169 65 110)(30 170 66 111)(31 171 67 112)(32 172 68 113)(33 173 69 114)(34 174 70 115)(35 175 71 116)(36 176 72 117)(49 208 230 122)(50 209 231 123)(51 210 232 124)(52 211 233 125)(53 212 234 126)(54 213 235 127)(55 214 236 128)(56 215 237 129)(57 216 238 130)(58 205 239 131)(59 206 240 132)(60 207 229 121)(73 88 204 143)(74 89 193 144)(75 90 194 133)(76 91 195 134)(77 92 196 135)(78 93 197 136)(79 94 198 137)(80 95 199 138)(81 96 200 139)(82 85 201 140)(83 86 202 141)(84 87 203 142)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 120)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 97)(36 98)(37 172)(38 173)(39 174)(40 175)(41 176)(42 177)(43 178)(44 179)(45 180)(46 169)(47 170)(48 171)(49 202)(50 203)(51 204)(52 193)(53 194)(54 195)(55 196)(56 197)(57 198)(58 199)(59 200)(60 201)(61 189)(62 190)(63 191)(64 192)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 232)(74 233)(75 234)(76 235)(77 236)(78 237)(79 238)(80 239)(81 240)(82 229)(83 230)(84 231)(85 207)(86 208)(87 209)(88 210)(89 211)(90 212)(91 213)(92 214)(93 215)(94 216)(95 205)(96 206)(121 140)(122 141)(123 142)(124 143)(125 144)(126 133)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)
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), (1,105,57,204,25)(2,106,58,193,26)(3,107,59,194,27)(4,108,60,195,28)(5,97,49,196,29)(6,98,50,197,30)(7,99,51,198,31)(8,100,52,199,32)(9,101,53,200,33)(10,102,54,201,34)(11,103,55,202,35)(12,104,56,203,36)(13,166,173,139,212)(14,167,174,140,213)(15,168,175,141,214)(16,157,176,142,215)(17,158,177,143,216)(18,159,178,144,205)(19,160,179,133,206)(20,161,180,134,207)(21,162,169,135,208)(22,163,170,136,209)(23,164,171,137,210)(24,165,172,138,211)(37,225,119,89,131)(38,226,120,90,132)(39,227,109,91,121)(40,228,110,92,122)(41,217,111,93,123)(42,218,112,94,124)(43,219,113,95,125)(44,220,114,96,126)(45,221,115,85,127)(46,222,116,86,128)(47,223,117,87,129)(48,224,118,88,130)(61,147,183,238,73)(62,148,184,239,74)(63,149,185,240,75)(64,150,186,229,76)(65,151,187,230,77)(66,152,188,231,78)(67,153,189,232,79)(68,154,190,233,80)(69,155,191,234,81)(70,156,192,235,82)(71,145,181,236,83)(72,146,182,237,84), (1,158,147,224)(2,159,148,225)(3,160,149,226)(4,161,150,227)(5,162,151,228)(6,163,152,217)(7,164,153,218)(8,165,154,219)(9,166,155,220)(10,167,156,221)(11,168,145,222)(12,157,146,223)(13,191,44,101)(14,192,45,102)(15,181,46,103)(16,182,47,104)(17,183,48,105)(18,184,37,106)(19,185,38,107)(20,186,39,108)(21,187,40,97)(22,188,41,98)(23,189,42,99)(24,190,43,100)(25,177,61,118)(26,178,62,119)(27,179,63,120)(28,180,64,109)(29,169,65,110)(30,170,66,111)(31,171,67,112)(32,172,68,113)(33,173,69,114)(34,174,70,115)(35,175,71,116)(36,176,72,117)(49,208,230,122)(50,209,231,123)(51,210,232,124)(52,211,233,125)(53,212,234,126)(54,213,235,127)(55,214,236,128)(56,215,237,129)(57,216,238,130)(58,205,239,131)(59,206,240,132)(60,207,229,121)(73,88,204,143)(74,89,193,144)(75,90,194,133)(76,91,195,134)(77,92,196,135)(78,93,197,136)(79,94,198,137)(80,95,199,138)(81,96,200,139)(82,85,201,140)(83,86,202,141)(84,87,203,142), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,172)(38,173)(39,174)(40,175)(41,176)(42,177)(43,178)(44,179)(45,180)(46,169)(47,170)(48,171)(49,202)(50,203)(51,204)(52,193)(53,194)(54,195)(55,196)(56,197)(57,198)(58,199)(59,200)(60,201)(61,189)(62,190)(63,191)(64,192)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,232)(74,233)(75,234)(76,235)(77,236)(78,237)(79,238)(80,239)(81,240)(82,229)(83,230)(84,231)(85,207)(86,208)(87,209)(88,210)(89,211)(90,212)(91,213)(92,214)(93,215)(94,216)(95,205)(96,206)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)>;
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), (1,105,57,204,25)(2,106,58,193,26)(3,107,59,194,27)(4,108,60,195,28)(5,97,49,196,29)(6,98,50,197,30)(7,99,51,198,31)(8,100,52,199,32)(9,101,53,200,33)(10,102,54,201,34)(11,103,55,202,35)(12,104,56,203,36)(13,166,173,139,212)(14,167,174,140,213)(15,168,175,141,214)(16,157,176,142,215)(17,158,177,143,216)(18,159,178,144,205)(19,160,179,133,206)(20,161,180,134,207)(21,162,169,135,208)(22,163,170,136,209)(23,164,171,137,210)(24,165,172,138,211)(37,225,119,89,131)(38,226,120,90,132)(39,227,109,91,121)(40,228,110,92,122)(41,217,111,93,123)(42,218,112,94,124)(43,219,113,95,125)(44,220,114,96,126)(45,221,115,85,127)(46,222,116,86,128)(47,223,117,87,129)(48,224,118,88,130)(61,147,183,238,73)(62,148,184,239,74)(63,149,185,240,75)(64,150,186,229,76)(65,151,187,230,77)(66,152,188,231,78)(67,153,189,232,79)(68,154,190,233,80)(69,155,191,234,81)(70,156,192,235,82)(71,145,181,236,83)(72,146,182,237,84), (1,158,147,224)(2,159,148,225)(3,160,149,226)(4,161,150,227)(5,162,151,228)(6,163,152,217)(7,164,153,218)(8,165,154,219)(9,166,155,220)(10,167,156,221)(11,168,145,222)(12,157,146,223)(13,191,44,101)(14,192,45,102)(15,181,46,103)(16,182,47,104)(17,183,48,105)(18,184,37,106)(19,185,38,107)(20,186,39,108)(21,187,40,97)(22,188,41,98)(23,189,42,99)(24,190,43,100)(25,177,61,118)(26,178,62,119)(27,179,63,120)(28,180,64,109)(29,169,65,110)(30,170,66,111)(31,171,67,112)(32,172,68,113)(33,173,69,114)(34,174,70,115)(35,175,71,116)(36,176,72,117)(49,208,230,122)(50,209,231,123)(51,210,232,124)(52,211,233,125)(53,212,234,126)(54,213,235,127)(55,214,236,128)(56,215,237,129)(57,216,238,130)(58,205,239,131)(59,206,240,132)(60,207,229,121)(73,88,204,143)(74,89,193,144)(75,90,194,133)(76,91,195,134)(77,92,196,135)(78,93,197,136)(79,94,198,137)(80,95,199,138)(81,96,200,139)(82,85,201,140)(83,86,202,141)(84,87,203,142), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,172)(38,173)(39,174)(40,175)(41,176)(42,177)(43,178)(44,179)(45,180)(46,169)(47,170)(48,171)(49,202)(50,203)(51,204)(52,193)(53,194)(54,195)(55,196)(56,197)(57,198)(58,199)(59,200)(60,201)(61,189)(62,190)(63,191)(64,192)(65,181)(66,182)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,232)(74,233)(75,234)(76,235)(77,236)(78,237)(79,238)(80,239)(81,240)(82,229)(83,230)(84,231)(85,207)(86,208)(87,209)(88,210)(89,211)(90,212)(91,213)(92,214)(93,215)(94,216)(95,205)(96,206)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228) );
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)], [(1,105,57,204,25),(2,106,58,193,26),(3,107,59,194,27),(4,108,60,195,28),(5,97,49,196,29),(6,98,50,197,30),(7,99,51,198,31),(8,100,52,199,32),(9,101,53,200,33),(10,102,54,201,34),(11,103,55,202,35),(12,104,56,203,36),(13,166,173,139,212),(14,167,174,140,213),(15,168,175,141,214),(16,157,176,142,215),(17,158,177,143,216),(18,159,178,144,205),(19,160,179,133,206),(20,161,180,134,207),(21,162,169,135,208),(22,163,170,136,209),(23,164,171,137,210),(24,165,172,138,211),(37,225,119,89,131),(38,226,120,90,132),(39,227,109,91,121),(40,228,110,92,122),(41,217,111,93,123),(42,218,112,94,124),(43,219,113,95,125),(44,220,114,96,126),(45,221,115,85,127),(46,222,116,86,128),(47,223,117,87,129),(48,224,118,88,130),(61,147,183,238,73),(62,148,184,239,74),(63,149,185,240,75),(64,150,186,229,76),(65,151,187,230,77),(66,152,188,231,78),(67,153,189,232,79),(68,154,190,233,80),(69,155,191,234,81),(70,156,192,235,82),(71,145,181,236,83),(72,146,182,237,84)], [(1,158,147,224),(2,159,148,225),(3,160,149,226),(4,161,150,227),(5,162,151,228),(6,163,152,217),(7,164,153,218),(8,165,154,219),(9,166,155,220),(10,167,156,221),(11,168,145,222),(12,157,146,223),(13,191,44,101),(14,192,45,102),(15,181,46,103),(16,182,47,104),(17,183,48,105),(18,184,37,106),(19,185,38,107),(20,186,39,108),(21,187,40,97),(22,188,41,98),(23,189,42,99),(24,190,43,100),(25,177,61,118),(26,178,62,119),(27,179,63,120),(28,180,64,109),(29,169,65,110),(30,170,66,111),(31,171,67,112),(32,172,68,113),(33,173,69,114),(34,174,70,115),(35,175,71,116),(36,176,72,117),(49,208,230,122),(50,209,231,123),(51,210,232,124),(52,211,233,125),(53,212,234,126),(54,213,235,127),(55,214,236,128),(56,215,237,129),(57,216,238,130),(58,205,239,131),(59,206,240,132),(60,207,229,121),(73,88,204,143),(74,89,193,144),(75,90,194,133),(76,91,195,134),(77,92,196,135),(78,93,197,136),(79,94,198,137),(80,95,199,138),(81,96,200,139),(82,85,201,140),(83,86,202,141),(84,87,203,142)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,120),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,97),(36,98),(37,172),(38,173),(39,174),(40,175),(41,176),(42,177),(43,178),(44,179),(45,180),(46,169),(47,170),(48,171),(49,202),(50,203),(51,204),(52,193),(53,194),(54,195),(55,196),(56,197),(57,198),(58,199),(59,200),(60,201),(61,189),(62,190),(63,191),(64,192),(65,181),(66,182),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(73,232),(74,233),(75,234),(76,235),(77,236),(78,237),(79,238),(80,239),(81,240),(82,229),(83,230),(84,231),(85,207),(86,208),(87,209),(88,210),(89,211),(90,212),(91,213),(92,214),(93,215),(94,216),(95,205),(96,206),(121,140),(122,141),(123,142),(124,143),(125,144),(126,133),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10N | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C4×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C5⋊D4 | D5×C12 | C3×C4○D20 |
| kernel | C12×C5⋊D4 | C12×Dic5 | C3×C10.D4 | C3×D10⋊C4 | C3×C23.D5 | D5×C2×C12 | C6×C5⋊D4 | C22×C60 | C4×C5⋊D4 | C3×C5⋊D4 | C4×Dic5 | C10.D4 | D10⋊C4 | C23.D5 | C2×C4×D5 | C2×C5⋊D4 | C22×C20 | C5⋊D4 | C60 | C22×C12 | C30 | C2×C12 | C22×C6 | C20 | C22×C4 | C12 | C2×C6 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C12×C5⋊D4 ►in GL3(𝔽61) generated by
| 32 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 17 | 60 |
| 0 | 1 | 0 |
| 60 | 0 | 0 |
| 0 | 14 | 45 |
| 0 | 39 | 47 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 17 | 60 |
G:=sub<GL(3,GF(61))| [32,0,0,0,1,0,0,0,1],[1,0,0,0,17,1,0,60,0],[60,0,0,0,14,39,0,45,47],[1,0,0,0,1,17,0,0,60] >;
C12×C5⋊D4 in GAP, Magma, Sage, TeX
C_{12}\times C_5\rtimes D_4 % in TeX
G:=Group("C12xC5:D4"); // GroupNames label
G:=SmallGroup(480,721);
// by ID
G=gap.SmallGroup(480,721);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,142,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations