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## G = C6×Q8⋊D5order 480 = 25·3·5

### Direct product of C6 and Q8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×Q8⋊D5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C6×D20 — C6×Q8⋊D5
 Lower central C5 — C10 — C20 — C6×Q8⋊D5
 Upper central C1 — C2×C6 — C2×C12 — C6×Q8

Generators and relations for C6×Q8⋊D5
G = < a,b,c,d,e | a6=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 496 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], D10 [×4], C2×C10, C24 [×2], C2×C12, C2×C12, C3×D4 [×3], C3×Q8 [×2], C3×Q8, C22×C6, C3×D5 [×2], C30, C30 [×2], C2×SD16, C52C8 [×2], D20 [×2], D20, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C22×D5, C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C60 [×2], C60 [×2], C6×D5 [×4], C2×C30, C2×C52C8, Q8⋊D5 [×4], C2×D20, Q8×C10, C6×SD16, C3×C52C8 [×2], C3×D20 [×2], C3×D20, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, D5×C2×C6, C2×Q8⋊D5, C6×C52C8, C3×Q8⋊D5 [×4], C6×D20, Q8×C30, C6×Q8⋊D5
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], SD16 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×SD16, C5⋊D4 [×2], C22×D5, C3×SD16 [×2], C6×D4, C6×D5 [×3], Q8⋊D5 [×2], C2×C5⋊D4, C6×SD16, C3×C5⋊D4 [×2], D5×C2×C6, C2×Q8⋊D5, C3×Q8⋊D5 [×2], C6×C5⋊D4, C6×Q8⋊D5

Smallest permutation representation of C6×Q8⋊D5
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)
(1 113 205 49)(2 114 206 50)(3 109 207 51)(4 110 208 52)(5 111 209 53)(6 112 210 54)(7 71 215 155)(8 72 216 156)(9 67 211 151)(10 68 212 152)(11 69 213 153)(12 70 214 154)(13 63 124 219)(14 64 125 220)(15 65 126 221)(16 66 121 222)(17 61 122 217)(18 62 123 218)(19 120 184 26)(20 115 185 27)(21 116 186 28)(22 117 181 29)(23 118 182 30)(24 119 183 25)(31 129 233 91)(32 130 234 92)(33 131 229 93)(34 132 230 94)(35 127 231 95)(36 128 232 96)(37 172 88 80)(38 173 89 81)(39 174 90 82)(40 169 85 83)(41 170 86 84)(42 171 87 79)(43 133 99 197)(44 134 100 198)(45 135 101 193)(46 136 102 194)(47 137 97 195)(48 138 98 196)(55 237 140 187)(56 238 141 188)(57 239 142 189)(58 240 143 190)(59 235 144 191)(60 236 139 192)(73 165 145 223)(74 166 146 224)(75 167 147 225)(76 168 148 226)(77 163 149 227)(78 164 150 228)(103 203 160 179)(104 204 161 180)(105 199 162 175)(106 200 157 176)(107 201 158 177)(108 202 159 178)
(1 200 205 176)(2 201 206 177)(3 202 207 178)(4 203 208 179)(5 204 209 180)(6 199 210 175)(7 19 215 184)(8 20 216 185)(9 21 211 186)(10 22 212 181)(11 23 213 182)(12 24 214 183)(13 146 124 74)(14 147 125 75)(15 148 126 76)(16 149 121 77)(17 150 122 78)(18 145 123 73)(25 154 119 70)(26 155 120 71)(27 156 115 72)(28 151 116 67)(29 152 117 68)(30 153 118 69)(31 37 233 88)(32 38 234 89)(33 39 229 90)(34 40 230 85)(35 41 231 86)(36 42 232 87)(43 143 99 58)(44 144 100 59)(45 139 101 60)(46 140 102 55)(47 141 97 56)(48 142 98 57)(49 157 113 106)(50 158 114 107)(51 159 109 108)(52 160 110 103)(53 161 111 104)(54 162 112 105)(61 164 217 228)(62 165 218 223)(63 166 219 224)(64 167 220 225)(65 168 221 226)(66 163 222 227)(79 96 171 128)(80 91 172 129)(81 92 173 130)(82 93 174 131)(83 94 169 132)(84 95 170 127)(133 240 197 190)(134 235 198 191)(135 236 193 192)(136 237 194 187)(137 238 195 188)(138 239 196 189)
(1 215 147 93 135)(2 216 148 94 136)(3 211 149 95 137)(4 212 150 96 138)(5 213 145 91 133)(6 214 146 92 134)(7 75 131 193 205)(8 76 132 194 206)(9 77 127 195 207)(10 78 128 196 208)(11 73 129 197 209)(12 74 130 198 210)(13 81 191 175 24)(14 82 192 176 19)(15 83 187 177 20)(16 84 188 178 21)(17 79 189 179 22)(18 80 190 180 23)(25 219 89 144 162)(26 220 90 139 157)(27 221 85 140 158)(28 222 86 141 159)(29 217 87 142 160)(30 218 88 143 161)(31 99 111 153 223)(32 100 112 154 224)(33 101 113 155 225)(34 102 114 156 226)(35 97 109 151 227)(36 98 110 152 228)(37 58 104 118 62)(38 59 105 119 63)(39 60 106 120 64)(40 55 107 115 65)(41 56 108 116 66)(42 57 103 117 61)(43 53 69 165 233)(44 54 70 166 234)(45 49 71 167 229)(46 50 72 168 230)(47 51 67 163 231)(48 52 68 164 232)(121 170 238 202 186)(122 171 239 203 181)(123 172 240 204 182)(124 173 235 199 183)(125 174 236 200 184)(126 169 237 201 185)
(1 138)(2 133)(3 134)(4 135)(5 136)(6 137)(7 128)(8 129)(9 130)(10 131)(11 132)(12 127)(13 66)(14 61)(15 62)(16 63)(17 64)(18 65)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 170)(26 171)(27 172)(28 173)(29 174)(30 169)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(43 114)(44 109)(45 110)(46 111)(47 112)(48 113)(49 98)(50 99)(51 100)(52 101)(53 102)(54 97)(55 180)(56 175)(57 176)(58 177)(59 178)(60 179)(73 76)(74 77)(75 78)(79 120)(80 115)(81 116)(82 117)(83 118)(84 119)(85 182)(86 183)(87 184)(88 185)(89 186)(90 181)(91 216)(92 211)(93 212)(94 213)(95 214)(96 215)(103 192)(104 187)(105 188)(106 189)(107 190)(108 191)(121 219)(122 220)(123 221)(124 222)(125 217)(126 218)(139 203)(140 204)(141 199)(142 200)(143 201)(144 202)(145 148)(146 149)(147 150)(151 234)(152 229)(153 230)(154 231)(155 232)(156 233)(157 239)(158 240)(159 235)(160 236)(161 237)(162 238)(163 224)(164 225)(165 226)(166 227)(167 228)(168 223)(193 208)(194 209)(195 210)(196 205)(197 206)(198 207)

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 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 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 SD16 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C3×SD16 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 Q8⋊D5 C3×Q8⋊D5 kernel C6×Q8⋊D5 C6×C5⋊2C8 C3×Q8⋊D5 C6×D20 Q8×C30 C2×Q8⋊D5 C2×C5⋊2C8 Q8⋊D5 C2×D20 Q8×C10 C60 C2×C30 C6×Q8 C30 C2×C12 C3×Q8 C20 C2×C10 C2×Q8 C12 C2×C6 C10 C2×C4 Q8 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 2 4 2 4 2 2 4 4 4 8 4 8 8 8 4 8

Matrix representation of C6×Q8⋊D5 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 226 0 0 0 0 226
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 0
,
 240 0 0 0 0 240 0 0 0 0 222 222 0 0 222 19
,
 240 240 0 0 53 52 0 0 0 0 1 0 0 0 0 1
,
 0 190 0 0 189 0 0 0 0 0 240 0 0 0 0 1
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,226,0,0,0,0,226],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,222,222,0,0,222,19],[240,53,0,0,240,52,0,0,0,0,1,0,0,0,0,1],[0,189,0,0,190,0,0,0,0,0,240,0,0,0,0,1] >;

C6×Q8⋊D5 in GAP, Magma, Sage, TeX

C_6\times Q_8\rtimes D_5
% in TeX

G:=Group("C6xQ8:D5");
// GroupNames label

G:=SmallGroup(480,734);
// by ID

G=gap.SmallGroup(480,734);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,268,2524,648,102,18822]);
// Polycyclic

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

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