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## G = C5×D4.D6order 480 = 25·3·5

### Direct product of C5 and D4.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D4.D6
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×Q8 — C5×D4.D6
 Lower central C3 — C6 — C12 — C5×D4.D6
 Upper central C1 — C10 — C20 — C5×SD16

Generators and relations for C5×D4.D6
G = < a,b,c,d,e | a5=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >

Subgroups: 276 in 120 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C10, C10 [×2], Dic3, Dic3 [×2], C12, C12, D6, C2×C6, C15, M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C20, C20 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6 [×2], Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C30, C30, C8.C22, C40, C40, C2×C20 [×3], C5×D4, C5×D4, C5×Q8, C5×Q8 [×3], C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C5×Dic3, C5×Dic3 [×2], C60, C60, S3×C10, C2×C30, C5×M4(2), C5×SD16, C5×SD16, C5×Q16 [×2], Q8×C10, C5×C4○D4, D4.D6, C5×C3⋊C8, C120, C5×Dic6 [×2], C5×Dic6, S3×C20, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, Q8×C15, C5×C8.C22, C5×C8⋊S3, C5×Dic12, C5×D4.S3, C5×C3⋊Q16, C15×SD16, C5×D42S3, C5×S3×Q8, C5×D4.D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C8.C22, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, D4.D6, S3×C2×C10, C5×C8.C22, C5×S3×D4, C5×D4.D6

Smallest permutation representation of C5×D4.D6
On 240 points
Generators in S240
(1 222 156 136 37)(2 223 145 137 38)(3 224 146 138 39)(4 225 147 139 40)(5 226 148 140 41)(6 227 149 141 42)(7 228 150 142 43)(8 217 151 143 44)(9 218 152 144 45)(10 219 153 133 46)(11 220 154 134 47)(12 221 155 135 48)(13 73 190 64 161)(14 74 191 65 162)(15 75 192 66 163)(16 76 181 67 164)(17 77 182 68 165)(18 78 183 69 166)(19 79 184 70 167)(20 80 185 71 168)(21 81 186 72 157)(22 82 187 61 158)(23 83 188 62 159)(24 84 189 63 160)(25 129 199 206 102)(26 130 200 207 103)(27 131 201 208 104)(28 132 202 209 105)(29 121 203 210 106)(30 122 204 211 107)(31 123 193 212 108)(32 124 194 213 97)(33 125 195 214 98)(34 126 196 215 99)(35 127 197 216 100)(36 128 198 205 101)(49 116 229 170 85)(50 117 230 171 86)(51 118 231 172 87)(52 119 232 173 88)(53 120 233 174 89)(54 109 234 175 90)(55 110 235 176 91)(56 111 236 177 92)(57 112 237 178 93)(58 113 238 179 94)(59 114 239 180 95)(60 115 240 169 96)
(1 34 7 28)(2 29 8 35)(3 36 9 30)(4 31 10 25)(5 26 11 32)(6 33 12 27)(13 90 19 96)(14 85 20 91)(15 92 21 86)(16 87 22 93)(17 94 23 88)(18 89 24 95)(37 99 43 105)(38 106 44 100)(39 101 45 107)(40 108 46 102)(41 103 47 97)(42 98 48 104)(49 80 55 74)(50 75 56 81)(51 82 57 76)(52 77 58 83)(53 84 59 78)(54 79 60 73)(61 237 67 231)(62 232 68 238)(63 239 69 233)(64 234 70 240)(65 229 71 235)(66 236 72 230)(109 184 115 190)(110 191 116 185)(111 186 117 192)(112 181 118 187)(113 188 119 182)(114 183 120 189)(121 217 127 223)(122 224 128 218)(123 219 129 225)(124 226 130 220)(125 221 131 227)(126 228 132 222)(133 206 139 212)(134 213 140 207)(135 208 141 214)(136 215 142 209)(137 210 143 216)(138 205 144 211)(145 203 151 197)(146 198 152 204)(147 193 153 199)(148 200 154 194)(149 195 155 201)(150 202 156 196)(157 171 163 177)(158 178 164 172)(159 173 165 179)(160 180 166 174)(161 175 167 169)(162 170 168 176)
(1 55)(2 81)(3 57)(4 83)(5 59)(6 73)(7 49)(8 75)(9 51)(10 77)(11 53)(12 79)(13 42)(14 105)(15 44)(16 107)(17 46)(18 97)(19 48)(20 99)(21 38)(22 101)(23 40)(24 103)(25 52)(26 84)(27 54)(28 74)(29 56)(30 76)(31 58)(32 78)(33 60)(34 80)(35 50)(36 82)(37 91)(39 93)(41 95)(43 85)(45 87)(47 89)(61 198)(62 147)(63 200)(64 149)(65 202)(66 151)(67 204)(68 153)(69 194)(70 155)(71 196)(72 145)(86 100)(88 102)(90 104)(92 106)(94 108)(96 98)(109 131)(110 222)(111 121)(112 224)(113 123)(114 226)(115 125)(116 228)(117 127)(118 218)(119 129)(120 220)(122 181)(124 183)(126 185)(128 187)(130 189)(132 191)(133 165)(134 174)(135 167)(136 176)(137 157)(138 178)(139 159)(140 180)(141 161)(142 170)(143 163)(144 172)(146 237)(148 239)(150 229)(152 231)(154 233)(156 235)(158 205)(160 207)(162 209)(164 211)(166 213)(168 215)(169 214)(171 216)(173 206)(175 208)(177 210)(179 212)(182 219)(184 221)(186 223)(188 225)(190 227)(192 217)(193 238)(195 240)(197 230)(199 232)(201 234)(203 236)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 42 43 48)(38 47 44 41)(39 40 45 46)(49 60 55 54)(50 53 56 59)(51 58 57 52)(61 68 67 62)(63 66 69 72)(64 71 70 65)(73 80 79 74)(75 78 81 84)(76 83 82 77)(85 96 91 90)(86 89 92 95)(87 94 93 88)(97 106 103 100)(98 99 104 105)(101 102 107 108)(109 116 115 110)(111 114 117 120)(112 119 118 113)(121 130 127 124)(122 123 128 129)(125 126 131 132)(133 138 139 144)(134 143 140 137)(135 136 141 142)(145 154 151 148)(146 147 152 153)(149 150 155 156)(157 160 163 166)(158 165 164 159)(161 168 167 162)(169 176 175 170)(171 174 177 180)(172 179 178 173)(181 188 187 182)(183 186 189 192)(184 191 190 185)(193 198 199 204)(194 203 200 197)(195 196 201 202)(205 206 211 212)(207 216 213 210)(208 209 214 215)(217 226 223 220)(218 219 224 225)(221 222 227 228)(229 240 235 234)(230 233 236 239)(231 238 237 232)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L 20M ··· 20T 24A 24B 30A 30B 30C 30D 30E 30F 30G 30H 40A 40B 40C 40D 40E 40F 40G 40H 60A 60B 60C 60D 60E 60F 60G 60H 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 ··· 20 24 24 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 120 ··· 120 size 1 1 4 6 2 2 4 6 12 12 1 1 1 1 2 8 4 12 1 1 1 1 4 4 4 4 6 6 6 6 4 8 2 2 2 2 2 2 2 2 4 4 4 4 6 6 6 6 12 ··· 12 4 4 2 2 2 2 8 8 8 8 4 4 4 4 12 12 12 12 4 4 4 4 8 8 8 8 4 ··· 4

90 irreducible representations

 dim 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 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D6 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C8.C22 S3×D4 D4.D6 C5×C8.C22 C5×S3×D4 C5×D4.D6 kernel C5×D4.D6 C5×C8⋊S3 C5×Dic12 C5×D4.S3 C5×C3⋊Q16 C15×SD16 C5×D4⋊2S3 C5×S3×Q8 D4.D6 C8⋊S3 Dic12 D4.S3 C3⋊Q16 C3×SD16 D4⋊2S3 S3×Q8 C5×SD16 C5×Dic3 S3×C10 C40 C5×D4 C5×Q8 SD16 Dic3 D6 C8 D4 Q8 C15 C10 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4 1 1 2 4 4 8

Matrix representation of C5×D4.D6 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 0 0 240 1 240 240 239 240 81 81 1 0 80 81 1 0
,
 87 213 73 73 101 227 0 73 112 157 14 28 140 112 140 154
,
 170 66 118 236 52 0 123 118 186 115 0 175 11 186 189 71
,
 52 0 123 118 170 66 118 236 186 115 0 175 63 115 52 123
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,240,81,80,0,240,81,81,240,239,1,1,1,240,0,0],[87,101,112,140,213,227,157,112,73,0,14,140,73,73,28,154],[170,52,186,11,66,0,115,186,118,123,0,189,236,118,175,71],[52,170,186,63,0,66,115,115,123,118,0,52,118,236,175,123] >;

C5×D4.D6 in GAP, Magma, Sage, TeX

C_5\times D_4.D_6
% in TeX

G:=Group("C5xD4.D6");
// GroupNames label

G:=SmallGroup(480,794);
// by ID

G=gap.SmallGroup(480,794);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,1766,471,2111,1068,102,15686]);
// Polycyclic

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

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