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## G = C5×D6⋊3Q8order 480 = 25·3·5

### Direct product of C5 and D6⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×D6⋊3Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C2×C20 — C5×D6⋊3Q8
 Lower central C3 — C2×C6 — C5×D6⋊3Q8
 Upper central C1 — C2×C10 — Q8×C10

Generators and relations for C5×D63Q8
G = < a,b,c,d,e | a5=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 324 in 148 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C22⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, D63Q8, S3×C20, C10×Dic3, C10×Dic3, C2×C60, C2×C60, Q8×C15, S3×C2×C10, C5×C22⋊Q8, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4, S3×C2×C20, Q8×C30, C5×D63Q8
Quotients:

Smallest permutation representation of C5×D63Q8
On 240 points
Generators in S240
(1 60 48 36 24)(2 55 43 31 19)(3 56 44 32 20)(4 57 45 33 21)(5 58 46 34 22)(6 59 47 35 23)(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 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(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 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 237)(8 236)(9 235)(10 240)(11 239)(12 238)(19 26)(20 25)(21 30)(22 29)(23 28)(24 27)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(67 77)(68 76)(69 75)(70 74)(71 73)(72 78)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)(91 101)(92 100)(93 99)(94 98)(95 97)(96 102)(103 113)(104 112)(105 111)(106 110)(107 109)(108 114)(115 125)(116 124)(117 123)(118 122)(119 121)(120 126)(127 134)(128 133)(129 138)(130 137)(131 136)(132 135)(139 146)(140 145)(141 150)(142 149)(143 148)(144 147)(151 158)(152 157)(153 162)(154 161)(155 160)(156 159)(163 170)(164 169)(165 174)(166 173)(167 172)(168 171)(175 182)(176 181)(177 186)(178 185)(179 184)(180 183)(187 197)(188 196)(189 195)(190 194)(191 193)(192 198)(199 209)(200 208)(201 207)(202 206)(203 205)(204 210)(211 221)(212 220)(213 219)(214 218)(215 217)(216 222)(223 233)(224 232)(225 231)(226 230)(227 229)(228 234)
(1 76 16 72)(2 77 17 67)(3 78 18 68)(4 73 13 69)(5 74 14 70)(6 75 15 71)(7 179 239 183)(8 180 240 184)(9 175 235 185)(10 176 236 186)(11 177 237 181)(12 178 238 182)(19 89 29 79)(20 90 30 80)(21 85 25 81)(22 86 26 82)(23 87 27 83)(24 88 28 84)(31 101 41 91)(32 102 42 92)(33 97 37 93)(34 98 38 94)(35 99 39 95)(36 100 40 96)(43 113 53 103)(44 114 54 104)(45 109 49 105)(46 110 50 106)(47 111 51 107)(48 112 52 108)(55 125 65 115)(56 126 66 116)(57 121 61 117)(58 122 62 118)(59 123 63 119)(60 124 64 120)(127 187 137 197)(128 188 138 198)(129 189 133 193)(130 190 134 194)(131 191 135 195)(132 192 136 196)(139 199 149 209)(140 200 150 210)(141 201 145 205)(142 202 146 206)(143 203 147 207)(144 204 148 208)(151 211 161 221)(152 212 162 222)(153 213 157 217)(154 214 158 218)(155 215 159 219)(156 216 160 220)(163 223 173 233)(164 224 174 234)(165 225 169 229)(166 226 170 230)(167 227 171 231)(168 228 172 232)
(1 136 16 132)(2 137 17 127)(3 138 18 128)(4 133 13 129)(5 134 14 130)(6 135 15 131)(7 123 239 119)(8 124 240 120)(9 125 235 115)(10 126 236 116)(11 121 237 117)(12 122 238 118)(19 149 29 139)(20 150 30 140)(21 145 25 141)(22 146 26 142)(23 147 27 143)(24 148 28 144)(31 161 41 151)(32 162 42 152)(33 157 37 153)(34 158 38 154)(35 159 39 155)(36 160 40 156)(43 173 53 163)(44 174 54 164)(45 169 49 165)(46 170 50 166)(47 171 51 167)(48 172 52 168)(55 185 65 175)(56 186 66 176)(57 181 61 177)(58 182 62 178)(59 183 63 179)(60 184 64 180)(67 197 77 187)(68 198 78 188)(69 193 73 189)(70 194 74 190)(71 195 75 191)(72 196 76 192)(79 209 89 199)(80 210 90 200)(81 205 85 201)(82 206 86 202)(83 207 87 203)(84 208 88 204)(91 221 101 211)(92 222 102 212)(93 217 97 213)(94 218 98 214)(95 219 99 215)(96 220 100 216)(103 233 113 223)(104 234 114 224)(105 229 109 225)(106 230 110 226)(107 231 111 227)(108 232 112 228)

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

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

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

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 C3⋊D4 C5×S3 C5×D4 C5×Q8 S3×C10 C5×C4○D4 C5×C3⋊D4 S3×Q8 Q8⋊3S3 C5×S3×Q8 C5×Q8⋊3S3 kernel C5×D6⋊3Q8 C5×Dic3⋊C4 C5×C4⋊Dic3 C5×D6⋊C4 S3×C2×C20 Q8×C30 D6⋊3Q8 Dic3⋊C4 C4⋊Dic3 D6⋊C4 S3×C2×C4 C6×Q8 Q8×C10 C60 S3×C10 C2×C20 C30 C20 C2×Q8 C12 D6 C2×C4 C6 C4 C10 C10 C2 C2 # reps 1 2 1 2 1 1 4 8 4 8 4 4 1 2 2 3 2 4 4 8 8 12 8 16 1 1 4 4

Matrix representation of C5×D63Q8 in GL4(𝔽61) generated by

 20 0 0 0 0 20 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 60 0 0 0 0 0 60 0 0 0 0 60
,
 52 43 0 0 18 9 0 0 0 0 60 40 0 0 3 1
,
 1 0 0 0 0 1 0 0 0 0 40 28 0 0 6 21
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,60,0,0,0,0,60],[52,18,0,0,43,9,0,0,0,0,60,3,0,0,40,1],[1,0,0,0,0,1,0,0,0,0,40,6,0,0,28,21] >;

C5×D63Q8 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes_3Q_8
% in TeX

G:=Group("C5xD6:3Q8");
// GroupNames label

G:=SmallGroup(480,825);
// by ID

G=gap.SmallGroup(480,825);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,436,15686]);
// 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=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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