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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 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, C2×C20, C2×C20 [×2], C2×C20 [×5], C5×Q8 [×2], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C5×Dic3 [×3], C60 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C22×C20, Q8×C10, D63Q8, S3×C20 [×2], C10×Dic3, C10×Dic3 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], S3×C2×C10, C5×C22⋊Q8, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×D6⋊C4 [×2], S3×C2×C20, Q8×C30, C5×D63Q8
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], C3⋊D4 [×2], C22×S3, C5×S3, C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, S3×Q8, Q83S3, C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8×C10, C5×C4○D4, D63Q8, C5×C3⋊D4 [×2], S3×C2×C10, C5×C22⋊Q8, C5×S3×Q8, C5×Q83S3, C10×C3⋊D4, C5×D63Q8

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

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

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

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

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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