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## G = C15×C22⋊Q8order 480 = 25·3·5

### Direct product of C15 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C22⋊Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — Q8×C30 — C15×C22⋊Q8
 Lower central C1 — C22 — C15×C22⋊Q8
 Upper central C1 — C2×C30 — C15×C22⋊Q8

Generators and relations for C15×C22⋊Q8
G = < a,b,c,d,e | a15=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 200 in 148 conjugacy classes, 96 normal (48 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], Q8 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×C10, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C22×C12, C6×Q8, C60 [×2], C60 [×5], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C22×C20, Q8×C10, C3×C22⋊Q8, C2×C60 [×2], C2×C60 [×4], C2×C60 [×2], Q8×C15 [×2], C22×C30, C5×C22⋊Q8, C15×C22⋊C4 [×2], C15×C4⋊C4, C15×C4⋊C4 [×2], C22×C60, Q8×C30, C15×C22⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], Q8 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C2×Q8, C4○D4, C2×C10 [×7], C3×D4 [×2], C3×Q8 [×2], C22×C6, C30 [×7], C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, C6×D4, C6×Q8, C3×C4○D4, C2×C30 [×7], D4×C10, Q8×C10, C5×C4○D4, C3×C22⋊Q8, D4×C15 [×2], Q8×C15 [×2], C22×C30, C5×C22⋊Q8, D4×C30, Q8×C30, C15×C4○D4, C15×C22⋊Q8

Smallest permutation representation of C15×C22⋊Q8
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)
(1 237)(2 238)(3 239)(4 240)(5 226)(6 227)(7 228)(8 229)(9 230)(10 231)(11 232)(12 233)(13 234)(14 235)(15 236)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 225)(23 211)(24 212)(25 213)(26 214)(27 215)(28 216)(29 217)(30 218)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 196)(72 197)(73 198)(74 199)(75 200)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 193)(92 194)(93 195)(94 181)(95 182)(96 183)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 192)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 209)(17 210)(18 196)(19 197)(20 198)(21 199)(22 200)(23 201)(24 202)(25 203)(26 204)(27 205)(28 206)(29 207)(30 208)(31 101)(32 102)(33 103)(34 104)(35 105)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(106 231)(107 232)(108 233)(109 234)(110 235)(111 236)(112 237)(113 238)(114 239)(115 240)(116 226)(117 227)(118 228)(119 229)(120 230)(166 194)(167 195)(168 181)(169 182)(170 183)(171 184)(172 185)(173 186)(174 187)(175 188)(176 189)(177 190)(178 191)(179 192)(180 193)
(1 82 237 151)(2 83 238 152)(3 84 239 153)(4 85 240 154)(5 86 226 155)(6 87 227 156)(7 88 228 157)(8 89 229 158)(9 90 230 159)(10 76 231 160)(11 77 232 161)(12 78 233 162)(13 79 234 163)(14 80 235 164)(15 81 236 165)(16 98 69 172)(17 99 70 173)(18 100 71 174)(19 101 72 175)(20 102 73 176)(21 103 74 177)(22 104 75 178)(23 105 61 179)(24 91 62 180)(25 92 63 166)(26 93 64 167)(27 94 65 168)(28 95 66 169)(29 96 67 170)(30 97 68 171)(31 222 188 197)(32 223 189 198)(33 224 190 199)(34 225 191 200)(35 211 192 201)(36 212 193 202)(37 213 194 203)(38 214 195 204)(39 215 181 205)(40 216 182 206)(41 217 183 207)(42 218 184 208)(43 219 185 209)(44 220 186 210)(45 221 187 196)(46 145 128 106)(47 146 129 107)(48 147 130 108)(49 148 131 109)(50 149 132 110)(51 150 133 111)(52 136 134 112)(53 137 135 113)(54 138 121 114)(55 139 122 115)(56 140 123 116)(57 141 124 117)(58 142 125 118)(59 143 126 119)(60 144 127 120)
(1 214 237 204)(2 215 238 205)(3 216 239 206)(4 217 240 207)(5 218 226 208)(6 219 227 209)(7 220 228 210)(8 221 229 196)(9 222 230 197)(10 223 231 198)(11 224 232 199)(12 225 233 200)(13 211 234 201)(14 212 235 202)(15 213 236 203)(16 141 69 117)(17 142 70 118)(18 143 71 119)(19 144 72 120)(20 145 73 106)(21 146 74 107)(22 147 75 108)(23 148 61 109)(24 149 62 110)(25 150 63 111)(26 136 64 112)(27 137 65 113)(28 138 66 114)(29 139 67 115)(30 140 68 116)(31 159 188 90)(32 160 189 76)(33 161 190 77)(34 162 191 78)(35 163 192 79)(36 164 193 80)(37 165 194 81)(38 151 195 82)(39 152 181 83)(40 153 182 84)(41 154 183 85)(42 155 184 86)(43 156 185 87)(44 157 186 88)(45 158 187 89)(46 176 128 102)(47 177 129 103)(48 178 130 104)(49 179 131 105)(50 180 132 91)(51 166 133 92)(52 167 134 93)(53 168 135 94)(54 169 121 95)(55 170 122 96)(56 171 123 97)(57 172 124 98)(58 173 125 99)(59 174 126 100)(60 175 127 101)

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

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

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

210 conjugacy classes

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

210 irreducible representations

 dim 1 1 1 1 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 type + + + + + + - image C1 C2 C2 C2 C2 C3 C5 C6 C6 C6 C6 C10 C10 C10 C10 C15 C30 C30 C30 C30 D4 Q8 C4○D4 C3×D4 C3×Q8 C5×D4 C5×Q8 C3×C4○D4 C5×C4○D4 D4×C15 Q8×C15 C15×C4○D4 kernel C15×C22⋊Q8 C15×C22⋊C4 C15×C4⋊C4 C22×C60 Q8×C30 C5×C22⋊Q8 C3×C22⋊Q8 C5×C22⋊C4 C5×C4⋊C4 C22×C20 Q8×C10 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C60 C2×C30 C30 C20 C2×C10 C12 C2×C6 C10 C6 C4 C22 C2 # reps 1 2 3 1 1 2 4 4 6 2 2 8 12 4 4 8 16 24 8 8 2 2 2 4 4 8 8 4 8 16 16 16

Matrix representation of C15×C22⋊Q8 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 22 0 0 0 0 22
,
 1 0 0 0 60 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 11 0 0 0 59 50
,
 60 59 0 0 0 1 0 0 0 0 40 13 0 0 27 21
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,22,0,0,0,0,22],[1,60,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,11,59,0,0,0,50],[60,0,0,0,59,1,0,0,0,0,40,27,0,0,13,21] >;

C15×C22⋊Q8 in GAP, Magma, Sage, TeX

C_{15}\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C15xC2^2:Q8");
// GroupNames label

G:=SmallGroup(480,927);
// by ID

G=gap.SmallGroup(480,927);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,1709,848,5126]);
// Polycyclic

G:=Group<a,b,c,d,e|a^15=b^2=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,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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