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## G = C6×D10⋊C4order 480 = 25·3·5

### Direct product of C6 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C22×C6 — C6×D10⋊C4
 Lower central C5 — C10 — C6×D10⋊C4
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C6×D10⋊C4
G = < a,b,c,d | a6=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 912 in 264 conjugacy classes, 114 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], C5, C6 [×3], C6 [×4], C6 [×4], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C12 [×4], C2×C6, C2×C6 [×6], C2×C6 [×16], C15, C22⋊C4 [×4], C22×C4, C22×C4, C24, Dic5 [×2], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C12 [×2], C2×C12 [×6], C22×C6, C22×C6 [×10], C3×D5 [×4], C30 [×3], C30 [×4], C2×C22⋊C4, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×6], C22×D5 [×4], C22×C10, C3×C22⋊C4 [×4], C22×C12, C22×C12, C23×C6, C3×Dic5 [×2], C60 [×2], C6×D5 [×4], C6×D5 [×12], C2×C30, C2×C30 [×6], D10⋊C4 [×4], C22×Dic5, C22×C20, C23×D5, C6×C22⋊C4, C6×Dic5 [×2], C6×Dic5 [×2], C2×C60 [×2], C2×C60 [×2], D5×C2×C6 [×6], D5×C2×C6 [×4], C22×C30, C2×D10⋊C4, C3×D10⋊C4 [×4], C2×C6×Dic5, C22×C60, D5×C22×C6, C6×D10⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, D5, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C12 [×6], C3×D4 [×4], C22×C6, C3×D5, C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C6×D5 [×3], D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C6×C22⋊C4, D5×C12 [×2], C3×D20 [×2], C3×C5⋊D4 [×2], D5×C2×C6, C2×D10⋊C4, C3×D10⋊C4 [×4], D5×C2×C12, C6×D20, C6×C5⋊D4, C6×D10⋊C4

Smallest permutation representation of C6×D10⋊C4
On 240 points
Generators in S240
(1 152 47 132 27 172)(2 153 48 133 28 173)(3 154 49 134 29 174)(4 155 50 135 30 175)(5 156 41 136 21 176)(6 157 42 137 22 177)(7 158 43 138 23 178)(8 159 44 139 24 179)(9 160 45 140 25 180)(10 151 46 131 26 171)(11 146 51 126 31 166)(12 147 52 127 32 167)(13 148 53 128 33 168)(14 149 54 129 34 169)(15 150 55 130 35 170)(16 141 56 121 36 161)(17 142 57 122 37 162)(18 143 58 123 38 163)(19 144 59 124 39 164)(20 145 60 125 40 165)(61 216 101 196 81 236)(62 217 102 197 82 237)(63 218 103 198 83 238)(64 219 104 199 84 239)(65 220 105 200 85 240)(66 211 106 191 86 231)(67 212 107 192 87 232)(68 213 108 193 88 233)(69 214 109 194 89 234)(70 215 110 195 90 235)(71 206 111 186 91 226)(72 207 112 187 92 227)(73 208 113 188 93 228)(74 209 114 189 94 229)(75 210 115 190 95 230)(76 201 116 181 96 221)(77 202 117 182 97 222)(78 203 118 183 98 223)(79 204 119 184 99 224)(80 205 120 185 100 225)
(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)
(1 126)(2 125)(3 124)(4 123)(5 122)(6 121)(7 130)(8 129)(9 128)(10 127)(11 132)(12 131)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 142)(22 141)(23 150)(24 149)(25 148)(26 147)(27 146)(28 145)(29 144)(30 143)(31 152)(32 151)(33 160)(34 159)(35 158)(36 157)(37 156)(38 155)(39 154)(40 153)(41 162)(42 161)(43 170)(44 169)(45 168)(46 167)(47 166)(48 165)(49 164)(50 163)(51 172)(52 171)(53 180)(54 179)(55 178)(56 177)(57 176)(58 175)(59 174)(60 173)(61 187)(62 186)(63 185)(64 184)(65 183)(66 182)(67 181)(68 190)(69 189)(70 188)(71 197)(72 196)(73 195)(74 194)(75 193)(76 192)(77 191)(78 200)(79 199)(80 198)(81 207)(82 206)(83 205)(84 204)(85 203)(86 202)(87 201)(88 210)(89 209)(90 208)(91 217)(92 216)(93 215)(94 214)(95 213)(96 212)(97 211)(98 220)(99 219)(100 218)(101 227)(102 226)(103 225)(104 224)(105 223)(106 222)(107 221)(108 230)(109 229)(110 228)(111 237)(112 236)(113 235)(114 234)(115 233)(116 232)(117 231)(118 240)(119 239)(120 238)
(1 77 17 67)(2 78 18 68)(3 79 19 69)(4 80 20 70)(5 71 11 61)(6 72 12 62)(7 73 13 63)(8 74 14 64)(9 75 15 65)(10 76 16 66)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)(121 191 131 181)(122 192 132 182)(123 193 133 183)(124 194 134 184)(125 195 135 185)(126 196 136 186)(127 197 137 187)(128 198 138 188)(129 199 139 189)(130 200 140 190)(141 211 151 201)(142 212 152 202)(143 213 153 203)(144 214 154 204)(145 215 155 205)(146 216 156 206)(147 217 157 207)(148 218 158 208)(149 219 159 209)(150 220 160 210)(161 231 171 221)(162 232 172 222)(163 233 173 223)(164 234 174 224)(165 235 175 225)(166 236 176 226)(167 237 177 227)(168 238 178 228)(169 239 179 229)(170 240 180 230)

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

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

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

156 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6N 6O ··· 6V 10A ··· 10N 12A ··· 12H 12I ··· 12P 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 10 10 10 10 1 1 2 2 2 2 10 10 10 10 2 2 1 ··· 1 10 ··· 10 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 D5 D10 D10 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C6×D5 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 kernel C6×D10⋊C4 C3×D10⋊C4 C2×C6×Dic5 C22×C60 D5×C22×C6 C2×D10⋊C4 D5×C2×C6 D10⋊C4 C22×Dic5 C22×C20 C23×D5 C22×D5 C2×C30 C22×C12 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 2 8 8 2 2 2 16 4 2 4 2 8 4 8 8 8 8 4 16 16 16

Matrix representation of C6×D10⋊C4 in GL4(𝔽61) generated by

 13 0 0 0 0 14 0 0 0 0 13 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 43 18 0 0 43 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 43 1
,
 11 0 0 0 0 1 0 0 0 0 31 44 0 0 17 30
G:=sub<GL(4,GF(61))| [13,0,0,0,0,14,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,43,43,0,0,18,1],[1,0,0,0,0,1,0,0,0,0,60,43,0,0,0,1],[11,0,0,0,0,1,0,0,0,0,31,17,0,0,44,30] >;

C6×D10⋊C4 in GAP, Magma, Sage, TeX

C_6\times D_{10}\rtimes C_4
% in TeX

G:=Group("C6xD10:C4");
// GroupNames label

G:=SmallGroup(480,720);
// by ID

G=gap.SmallGroup(480,720);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,142,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

׿
×
𝔽