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## G = C12×C5⋊D4order 480 = 25·3·5

### Direct product of C12 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C12×C5⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×C5⋊D4 — C12×C5⋊D4
 Lower central C5 — C10 — C12×C5⋊D4
 Upper central C1 — C2×C12 — C22×C12

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 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C12 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×7], C3×D4 [×4], C22×C6, C22×C6, C3×D5 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C3×Dic5 [×2], C3×Dic5 [×2], C60 [×2], C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, D4×C12, D5×C12 [×2], C6×Dic5 [×3], C3×C5⋊D4 [×4], C2×C60 [×2], C2×C60 [×2], 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 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, D10 [×3], C2×C12 [×6], C3×D4 [×2], C22×C6, C3×D5, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5 [×3], C2×C4×D5, C4○D20, C2×C5⋊D4, D4×C12, D5×C12 [×2], C3×C5⋊D4 [×2], D5×C2×C6, C4×C5⋊D4, D5×C2×C12, C3×C4○D20, C6×C5⋊D4, C12×C5⋊D4

Smallest permutation representation of 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 83 89 128 21)(2 84 90 129 22)(3 73 91 130 23)(4 74 92 131 24)(5 75 93 132 13)(6 76 94 121 14)(7 77 95 122 15)(8 78 96 123 16)(9 79 85 124 17)(10 80 86 125 18)(11 81 87 126 19)(12 82 88 127 20)(25 193 117 106 136)(26 194 118 107 137)(27 195 119 108 138)(28 196 120 97 139)(29 197 109 98 140)(30 198 110 99 141)(31 199 111 100 142)(32 200 112 101 143)(33 201 113 102 144)(34 202 114 103 133)(35 203 115 104 134)(36 204 116 105 135)(37 219 157 59 209)(38 220 158 60 210)(39 221 159 49 211)(40 222 160 50 212)(41 223 161 51 213)(42 224 162 52 214)(43 225 163 53 215)(44 226 164 54 216)(45 227 165 55 205)(46 228 166 56 206)(47 217 167 57 207)(48 218 168 58 208)(61 147 181 234 175)(62 148 182 235 176)(63 149 183 236 177)(64 150 184 237 178)(65 151 185 238 179)(66 152 186 239 180)(67 153 187 240 169)(68 154 188 229 170)(69 155 189 230 171)(70 156 190 231 172)(71 145 191 232 173)(72 146 192 233 174)
(1 238 100 164)(2 239 101 165)(3 240 102 166)(4 229 103 167)(5 230 104 168)(6 231 105 157)(7 232 106 158)(8 233 107 159)(9 234 108 160)(10 235 97 161)(11 236 98 162)(12 237 99 163)(13 171 115 58)(14 172 116 59)(15 173 117 60)(16 174 118 49)(17 175 119 50)(18 176 120 51)(19 177 109 52)(20 178 110 53)(21 179 111 54)(22 180 112 55)(23 169 113 56)(24 170 114 57)(25 38 95 145)(26 39 96 146)(27 40 85 147)(28 41 86 148)(29 42 87 149)(30 43 88 150)(31 44 89 151)(32 45 90 152)(33 46 91 153)(34 47 92 154)(35 48 93 155)(36 37 94 156)(61 195 212 124)(62 196 213 125)(63 197 214 126)(64 198 215 127)(65 199 216 128)(66 200 205 129)(67 201 206 130)(68 202 207 131)(69 203 208 132)(70 204 209 121)(71 193 210 122)(72 194 211 123)(73 187 144 228)(74 188 133 217)(75 189 134 218)(76 190 135 219)(77 191 136 220)(78 192 137 221)(79 181 138 222)(80 182 139 223)(81 183 140 224)(82 184 141 225)(83 185 142 226)(84 186 143 227)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 81)(14 82)(15 83)(16 84)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 199)(26 200)(27 201)(28 202)(29 203)(30 204)(31 193)(32 194)(33 195)(34 196)(35 197)(36 198)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)(49 186)(50 187)(51 188)(52 189)(53 190)(54 191)(55 192)(56 181)(57 182)(58 183)(59 184)(60 185)(85 130)(86 131)(87 132)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 133)(145 216)(146 205)(147 206)(148 207)(149 208)(150 209)(151 210)(152 211)(153 212)(154 213)(155 214)(156 215)(157 237)(158 238)(159 239)(160 240)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)(169 222)(170 223)(171 224)(172 225)(173 226)(174 227)(175 228)(176 217)(177 218)(178 219)(179 220)(180 221)

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), (1,83,89,128,21)(2,84,90,129,22)(3,73,91,130,23)(4,74,92,131,24)(5,75,93,132,13)(6,76,94,121,14)(7,77,95,122,15)(8,78,96,123,16)(9,79,85,124,17)(10,80,86,125,18)(11,81,87,126,19)(12,82,88,127,20)(25,193,117,106,136)(26,194,118,107,137)(27,195,119,108,138)(28,196,120,97,139)(29,197,109,98,140)(30,198,110,99,141)(31,199,111,100,142)(32,200,112,101,143)(33,201,113,102,144)(34,202,114,103,133)(35,203,115,104,134)(36,204,116,105,135)(37,219,157,59,209)(38,220,158,60,210)(39,221,159,49,211)(40,222,160,50,212)(41,223,161,51,213)(42,224,162,52,214)(43,225,163,53,215)(44,226,164,54,216)(45,227,165,55,205)(46,228,166,56,206)(47,217,167,57,207)(48,218,168,58,208)(61,147,181,234,175)(62,148,182,235,176)(63,149,183,236,177)(64,150,184,237,178)(65,151,185,238,179)(66,152,186,239,180)(67,153,187,240,169)(68,154,188,229,170)(69,155,189,230,171)(70,156,190,231,172)(71,145,191,232,173)(72,146,192,233,174), (1,238,100,164)(2,239,101,165)(3,240,102,166)(4,229,103,167)(5,230,104,168)(6,231,105,157)(7,232,106,158)(8,233,107,159)(9,234,108,160)(10,235,97,161)(11,236,98,162)(12,237,99,163)(13,171,115,58)(14,172,116,59)(15,173,117,60)(16,174,118,49)(17,175,119,50)(18,176,120,51)(19,177,109,52)(20,178,110,53)(21,179,111,54)(22,180,112,55)(23,169,113,56)(24,170,114,57)(25,38,95,145)(26,39,96,146)(27,40,85,147)(28,41,86,148)(29,42,87,149)(30,43,88,150)(31,44,89,151)(32,45,90,152)(33,46,91,153)(34,47,92,154)(35,48,93,155)(36,37,94,156)(61,195,212,124)(62,196,213,125)(63,197,214,126)(64,198,215,127)(65,199,216,128)(66,200,205,129)(67,201,206,130)(68,202,207,131)(69,203,208,132)(70,204,209,121)(71,193,210,122)(72,194,211,123)(73,187,144,228)(74,188,133,217)(75,189,134,218)(76,190,135,219)(77,191,136,220)(78,192,137,221)(79,181,138,222)(80,182,139,223)(81,183,140,224)(82,184,141,225)(83,185,142,226)(84,186,143,227), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,81)(14,82)(15,83)(16,84)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,199)(26,200)(27,201)(28,202)(29,203)(30,204)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63)(49,186)(50,187)(51,188)(52,189)(53,190)(54,191)(55,192)(56,181)(57,182)(58,183)(59,184)(60,185)(85,130)(86,131)(87,132)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,133)(145,216)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,237)(158,238)(159,239)(160,240)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,222)(170,223)(171,224)(172,225)(173,226)(174,227)(175,228)(176,217)(177,218)(178,219)(179,220)(180,221)>;

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), (1,83,89,128,21)(2,84,90,129,22)(3,73,91,130,23)(4,74,92,131,24)(5,75,93,132,13)(6,76,94,121,14)(7,77,95,122,15)(8,78,96,123,16)(9,79,85,124,17)(10,80,86,125,18)(11,81,87,126,19)(12,82,88,127,20)(25,193,117,106,136)(26,194,118,107,137)(27,195,119,108,138)(28,196,120,97,139)(29,197,109,98,140)(30,198,110,99,141)(31,199,111,100,142)(32,200,112,101,143)(33,201,113,102,144)(34,202,114,103,133)(35,203,115,104,134)(36,204,116,105,135)(37,219,157,59,209)(38,220,158,60,210)(39,221,159,49,211)(40,222,160,50,212)(41,223,161,51,213)(42,224,162,52,214)(43,225,163,53,215)(44,226,164,54,216)(45,227,165,55,205)(46,228,166,56,206)(47,217,167,57,207)(48,218,168,58,208)(61,147,181,234,175)(62,148,182,235,176)(63,149,183,236,177)(64,150,184,237,178)(65,151,185,238,179)(66,152,186,239,180)(67,153,187,240,169)(68,154,188,229,170)(69,155,189,230,171)(70,156,190,231,172)(71,145,191,232,173)(72,146,192,233,174), (1,238,100,164)(2,239,101,165)(3,240,102,166)(4,229,103,167)(5,230,104,168)(6,231,105,157)(7,232,106,158)(8,233,107,159)(9,234,108,160)(10,235,97,161)(11,236,98,162)(12,237,99,163)(13,171,115,58)(14,172,116,59)(15,173,117,60)(16,174,118,49)(17,175,119,50)(18,176,120,51)(19,177,109,52)(20,178,110,53)(21,179,111,54)(22,180,112,55)(23,169,113,56)(24,170,114,57)(25,38,95,145)(26,39,96,146)(27,40,85,147)(28,41,86,148)(29,42,87,149)(30,43,88,150)(31,44,89,151)(32,45,90,152)(33,46,91,153)(34,47,92,154)(35,48,93,155)(36,37,94,156)(61,195,212,124)(62,196,213,125)(63,197,214,126)(64,198,215,127)(65,199,216,128)(66,200,205,129)(67,201,206,130)(68,202,207,131)(69,203,208,132)(70,204,209,121)(71,193,210,122)(72,194,211,123)(73,187,144,228)(74,188,133,217)(75,189,134,218)(76,190,135,219)(77,191,136,220)(78,192,137,221)(79,181,138,222)(80,182,139,223)(81,183,140,224)(82,184,141,225)(83,185,142,226)(84,186,143,227), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,81)(14,82)(15,83)(16,84)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,199)(26,200)(27,201)(28,202)(29,203)(30,204)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63)(49,186)(50,187)(51,188)(52,189)(53,190)(54,191)(55,192)(56,181)(57,182)(58,183)(59,184)(60,185)(85,130)(86,131)(87,132)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,133)(145,216)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,237)(158,238)(159,239)(160,240)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,222)(170,223)(171,224)(172,225)(173,226)(174,227)(175,228)(176,217)(177,218)(178,219)(179,220)(180,221) );

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)], [(1,83,89,128,21),(2,84,90,129,22),(3,73,91,130,23),(4,74,92,131,24),(5,75,93,132,13),(6,76,94,121,14),(7,77,95,122,15),(8,78,96,123,16),(9,79,85,124,17),(10,80,86,125,18),(11,81,87,126,19),(12,82,88,127,20),(25,193,117,106,136),(26,194,118,107,137),(27,195,119,108,138),(28,196,120,97,139),(29,197,109,98,140),(30,198,110,99,141),(31,199,111,100,142),(32,200,112,101,143),(33,201,113,102,144),(34,202,114,103,133),(35,203,115,104,134),(36,204,116,105,135),(37,219,157,59,209),(38,220,158,60,210),(39,221,159,49,211),(40,222,160,50,212),(41,223,161,51,213),(42,224,162,52,214),(43,225,163,53,215),(44,226,164,54,216),(45,227,165,55,205),(46,228,166,56,206),(47,217,167,57,207),(48,218,168,58,208),(61,147,181,234,175),(62,148,182,235,176),(63,149,183,236,177),(64,150,184,237,178),(65,151,185,238,179),(66,152,186,239,180),(67,153,187,240,169),(68,154,188,229,170),(69,155,189,230,171),(70,156,190,231,172),(71,145,191,232,173),(72,146,192,233,174)], [(1,238,100,164),(2,239,101,165),(3,240,102,166),(4,229,103,167),(5,230,104,168),(6,231,105,157),(7,232,106,158),(8,233,107,159),(9,234,108,160),(10,235,97,161),(11,236,98,162),(12,237,99,163),(13,171,115,58),(14,172,116,59),(15,173,117,60),(16,174,118,49),(17,175,119,50),(18,176,120,51),(19,177,109,52),(20,178,110,53),(21,179,111,54),(22,180,112,55),(23,169,113,56),(24,170,114,57),(25,38,95,145),(26,39,96,146),(27,40,85,147),(28,41,86,148),(29,42,87,149),(30,43,88,150),(31,44,89,151),(32,45,90,152),(33,46,91,153),(34,47,92,154),(35,48,93,155),(36,37,94,156),(61,195,212,124),(62,196,213,125),(63,197,214,126),(64,198,215,127),(65,199,216,128),(66,200,205,129),(67,201,206,130),(68,202,207,131),(69,203,208,132),(70,204,209,121),(71,193,210,122),(72,194,211,123),(73,187,144,228),(74,188,133,217),(75,189,134,218),(76,190,135,219),(77,191,136,220),(78,192,137,221),(79,181,138,222),(80,182,139,223),(81,183,140,224),(82,184,141,225),(83,185,142,226),(84,186,143,227)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,81),(14,82),(15,83),(16,84),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,199),(26,200),(27,201),(28,202),(29,203),(30,204),(31,193),(32,194),(33,195),(34,196),(35,197),(36,198),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,61),(47,62),(48,63),(49,186),(50,187),(51,188),(52,189),(53,190),(54,191),(55,192),(56,181),(57,182),(58,183),(59,184),(60,185),(85,130),(86,131),(87,132),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,133),(145,216),(146,205),(147,206),(148,207),(149,208),(150,209),(151,210),(152,211),(153,212),(154,213),(155,214),(156,215),(157,237),(158,238),(159,239),(160,240),(161,229),(162,230),(163,231),(164,232),(165,233),(166,234),(167,235),(168,236),(169,222),(170,223),(171,224),(172,225),(173,226),(174,227),(175,228),(176,217),(177,218),(178,219),(179,220),(180,221)])

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

׿
×
𝔽