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## G = C5×D6⋊Q8order 480 = 25·3·5

### Direct product of C5 and D6⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×D6⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C2×C20 — C5×D6⋊Q8
 Lower central C3 — C2×C6 — C5×D6⋊Q8
 Upper central C1 — C2×C10 — C5×C4⋊C4

Generators and relations for C5×D6⋊Q8
G = < a,b,c,d,e | a5=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d-1 >

Subgroups: 340 in 148 conjugacy classes, 66 normal (58 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C20 [×7], C2×C10, C2×C10 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×C12 [×3], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, C2×C20 [×3], C2×C20 [×5], C5×Q8 [×2], C22×C10, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C5×Dic3 [×2], C60 [×3], S3×C10 [×2], S3×C10 [×2], C2×C30, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C22×C20, Q8×C10, D6⋊Q8, C5×Dic6 [×2], S3×C20 [×2], C10×Dic3 [×3], C2×C60 [×3], S3×C2×C10, C5×C22⋊Q8, C5×Dic3⋊C4 [×2], C5×D6⋊C4 [×2], C15×C4⋊C4, C10×Dic6, S3×C2×C20, C5×D6⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], C22×S3, C5×S3, C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, C4○D12, S3×D4, S3×Q8, S3×C10 [×3], D4×C10, Q8×C10, C5×C4○D4, D6⋊Q8, S3×C2×C10, C5×C22⋊Q8, C5×C4○D12, C5×S3×D4, C5×S3×Q8, C5×D6⋊Q8

Smallest permutation representation of C5×D6⋊Q8
On 240 points
Generators in S240
(1 61 49 37 25)(2 62 50 38 26)(3 63 51 39 27)(4 64 52 40 28)(5 65 53 41 29)(6 66 54 42 30)(7 231 219 207 195)(8 232 220 208 196)(9 233 221 209 197)(10 234 222 210 198)(11 229 217 205 193)(12 230 218 206 194)(13 60 48 36 24)(14 55 43 31 19)(15 56 44 32 20)(16 57 45 33 21)(17 58 46 34 22)(18 59 47 35 23)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 121 109 97 85)(74 122 110 98 86)(75 123 111 99 87)(76 124 112 100 88)(77 125 113 101 89)(78 126 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 236)(8 235)(9 240)(10 239)(11 238)(12 237)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)(43 49)(44 54)(45 53)(46 52)(47 51)(48 50)(55 61)(56 66)(57 65)(58 64)(59 63)(60 62)(67 76)(68 75)(69 74)(70 73)(71 78)(72 77)(79 88)(80 87)(81 86)(82 85)(83 90)(84 89)(91 100)(92 99)(93 98)(94 97)(95 102)(96 101)(103 112)(104 111)(105 110)(106 109)(107 114)(108 113)(115 124)(116 123)(117 122)(118 121)(119 126)(120 125)(127 133)(128 138)(129 137)(130 136)(131 135)(132 134)(139 145)(140 150)(141 149)(142 148)(143 147)(144 146)(151 157)(152 162)(153 161)(154 160)(155 159)(156 158)(163 169)(164 174)(165 173)(166 172)(167 171)(168 170)(175 181)(176 186)(177 185)(178 184)(179 183)(180 182)(187 196)(188 195)(189 194)(190 193)(191 198)(192 197)(199 208)(200 207)(201 206)(202 205)(203 210)(204 209)(211 220)(212 219)(213 218)(214 217)(215 222)(216 221)(223 232)(224 231)(225 230)(226 229)(227 234)(228 233)
(1 76 18 68)(2 75 13 67)(3 74 14 72)(4 73 15 71)(5 78 16 70)(6 77 17 69)(7 185 235 177)(8 184 236 176)(9 183 237 175)(10 182 238 180)(11 181 239 179)(12 186 240 178)(19 84 27 86)(20 83 28 85)(21 82 29 90)(22 81 30 89)(23 80 25 88)(24 79 26 87)(31 96 39 98)(32 95 40 97)(33 94 41 102)(34 93 42 101)(35 92 37 100)(36 91 38 99)(43 108 51 110)(44 107 52 109)(45 106 53 114)(46 105 54 113)(47 104 49 112)(48 103 50 111)(55 120 63 122)(56 119 64 121)(57 118 65 126)(58 117 66 125)(59 116 61 124)(60 115 62 123)(127 197 135 189)(128 196 136 188)(129 195 137 187)(130 194 138 192)(131 193 133 191)(132 198 134 190)(139 209 147 201)(140 208 148 200)(141 207 149 199)(142 206 150 204)(143 205 145 203)(144 210 146 202)(151 221 159 213)(152 220 160 212)(153 219 161 211)(154 218 162 216)(155 217 157 215)(156 222 158 214)(163 233 171 225)(164 232 172 224)(165 231 173 223)(166 230 174 228)(167 229 169 227)(168 234 170 226)
(1 136 18 128)(2 135 13 127)(3 134 14 132)(4 133 15 131)(5 138 16 130)(6 137 17 129)(7 117 235 125)(8 116 236 124)(9 115 237 123)(10 120 238 122)(11 119 239 121)(12 118 240 126)(19 144 27 146)(20 143 28 145)(21 142 29 150)(22 141 30 149)(23 140 25 148)(24 139 26 147)(31 156 39 158)(32 155 40 157)(33 154 41 162)(34 153 42 161)(35 152 37 160)(36 151 38 159)(43 168 51 170)(44 167 52 169)(45 166 53 174)(46 165 54 173)(47 164 49 172)(48 163 50 171)(55 180 63 182)(56 179 64 181)(57 178 65 186)(58 177 66 185)(59 176 61 184)(60 175 62 183)(67 189 75 197)(68 188 76 196)(69 187 77 195)(70 192 78 194)(71 191 73 193)(72 190 74 198)(79 201 87 209)(80 200 88 208)(81 199 89 207)(82 204 90 206)(83 203 85 205)(84 202 86 210)(91 213 99 221)(92 212 100 220)(93 211 101 219)(94 216 102 218)(95 215 97 217)(96 214 98 222)(103 225 111 233)(104 224 112 232)(105 223 113 231)(106 228 114 230)(107 227 109 229)(108 226 110 234)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10T 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20X 20Y ··· 20AF 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 2 2 2 4 4 6 6 12 12 1 1 1 1 2 2 2 1 ··· 1 6 ··· 6 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 6 ··· 6 12 ··· 12 2 ··· 2 4 ··· 4

120 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 4 4 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 Q8 D6 C4○D4 C5×S3 C5×D4 C5×Q8 C4○D12 S3×C10 C5×C4○D4 C5×C4○D12 S3×D4 S3×Q8 C5×S3×D4 C5×S3×Q8 kernel C5×D6⋊Q8 C5×Dic3⋊C4 C5×D6⋊C4 C15×C4⋊C4 C10×Dic6 S3×C2×C20 D6⋊Q8 Dic3⋊C4 D6⋊C4 C3×C4⋊C4 C2×Dic6 S3×C2×C4 C5×C4⋊C4 C5×Dic3 S3×C10 C2×C20 C30 C4⋊C4 Dic3 D6 C10 C2×C4 C6 C2 C10 C10 C2 C2 # reps 1 2 2 1 1 1 4 8 8 4 4 4 1 2 2 3 2 4 8 8 4 12 8 16 1 1 4 4

Matrix representation of C5×D6⋊Q8 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 20 0 0 0 0 20
,
 0 60 0 0 1 1 0 0 0 0 60 0 0 0 0 60
,
 0 60 0 0 60 0 0 0 0 0 1 0 0 0 0 60
,
 52 43 0 0 52 9 0 0 0 0 0 60 0 0 60 0
,
 11 0 0 0 50 50 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,20,0,0,0,0,20],[0,1,0,0,60,1,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,0,60],[52,52,0,0,43,9,0,0,0,0,0,60,0,0,60,0],[11,50,0,0,0,50,0,0,0,0,60,0,0,0,0,60] >;

C5×D6⋊Q8 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes Q_8
% in TeX

G:=Group("C5xD6:Q8");
// GroupNames label

G:=SmallGroup(480,775);
// by ID

G=gap.SmallGroup(480,775);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,288,2606,891,436,15686]);
// Polycyclic

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

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