direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C22⋊Q8, C60.247D4, C4⋊C4⋊3C30, (C2×C30)⋊8Q8, (C6×Q8)⋊8C10, (C2×Q8)⋊3C30, C2.6(D4×C30), C2.3(Q8×C30), (Q8×C30)⋊22C2, (Q8×C10)⋊12C6, C4.13(D4×C15), C6.69(D4×C10), C12.62(C5×D4), C20.62(C3×D4), C10.69(C6×D4), C10.20(C6×Q8), C6.20(Q8×C10), C22⋊2(Q8×C15), C30.452(C2×D4), C22⋊C4.1C30, (C22×C4).7C30, C30.118(C2×Q8), C23.11(C2×C30), (C22×C60).35C2, (C22×C20).19C6, C30.278(C4○D4), (C22×C12).15C10, (C2×C30).457C23, (C2×C60).579C22, C22.12(C22×C30), (C22×C30).132C22, (C5×C4⋊C4)⋊12C6, (C2×C6)⋊2(C5×Q8), (C3×C4⋊C4)⋊12C10, (C15×C4⋊C4)⋊30C2, (C2×C10)⋊4(C3×Q8), (C2×C4).4(C2×C30), C2.5(C15×C4○D4), C6.42(C5×C4○D4), (C2×C20).66(C2×C6), C10.42(C3×C4○D4), (C5×C22⋊C4).4C6, (C2×C12).66(C2×C10), (C3×C22⋊C4).4C10, (C15×C22⋊C4).10C2, (C2×C6).77(C22×C10), (C2×C10).77(C22×C6), (C22×C6).28(C2×C10), (C22×C10).36(C2×C6), SmallGroup(480,927)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C30, C30, C22⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C60, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C3×C22⋊Q8, C2×C60, C2×C60, C2×C60, Q8×C15, C22×C30, C5×C22⋊Q8, C15×C22⋊C4, C15×C4⋊C4, C15×C4⋊C4, C22×C60, Q8×C30, C15×C22⋊Q8
Quotients: C1, C2, C3, C22, C5, C6, D4, Q8, C23, C10, C2×C6, C15, C2×D4, C2×Q8, C4○D4, C2×C10, C3×D4, C3×Q8, C22×C6, C30, C22⋊Q8, C5×D4, C5×Q8, C22×C10, C6×D4, C6×Q8, C3×C4○D4, C2×C30, D4×C10, Q8×C10, C5×C4○D4, C3×C22⋊Q8, D4×C15, Q8×C15, C22×C30, C5×C22⋊Q8, D4×C30, Q8×C30, C15×C4○D4, 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)
(1 189)(2 190)(3 191)(4 192)(5 193)(6 194)(7 195)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 106)(27 107)(28 108)(29 109)(30 110)(31 221)(32 222)(33 223)(34 224)(35 225)(36 211)(37 212)(38 213)(39 214)(40 215)(41 216)(42 217)(43 218)(44 219)(45 220)(46 135)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 166)(88 167)(89 168)(90 169)(91 204)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 196)(99 197)(100 198)(101 199)(102 200)(103 201)(104 202)(105 203)(151 232)(152 233)(153 234)(154 235)(155 236)(156 237)(157 238)(158 239)(159 240)(160 226)(161 227)(162 228)(163 229)(164 230)(165 231)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 16)(14 17)(15 18)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 46)(42 47)(43 48)(44 49)(45 50)(61 165)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 161)(73 162)(74 163)(75 164)(76 104)(77 105)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 217)(122 218)(123 219)(124 220)(125 221)(126 222)(127 223)(128 224)(129 225)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 240)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 233)(145 234)(146 235)(147 236)(148 237)(149 238)(150 239)(166 198)(167 199)(168 200)(169 201)(170 202)(171 203)(172 204)(173 205)(174 206)(175 207)(176 208)(177 209)(178 210)(179 196)(180 197)
(1 162 189 228)(2 163 190 229)(3 164 191 230)(4 165 192 231)(5 151 193 232)(6 152 194 233)(7 153 195 234)(8 154 181 235)(9 155 182 236)(10 156 183 237)(11 157 184 238)(12 158 185 239)(13 159 186 240)(14 160 187 226)(15 161 188 227)(16 70 111 136)(17 71 112 137)(18 72 113 138)(19 73 114 139)(20 74 115 140)(21 75 116 141)(22 61 117 142)(23 62 118 143)(24 63 119 144)(25 64 120 145)(26 65 106 146)(27 66 107 147)(28 67 108 148)(29 68 109 149)(30 69 110 150)(31 170 125 104)(32 171 126 105)(33 172 127 91)(34 173 128 92)(35 174 129 93)(36 175 130 94)(37 176 131 95)(38 177 132 96)(39 178 133 97)(40 179 134 98)(41 180 135 99)(42 166 121 100)(43 167 122 101)(44 168 123 102)(45 169 124 103)(46 197 216 86)(47 198 217 87)(48 199 218 88)(49 200 219 89)(50 201 220 90)(51 202 221 76)(52 203 222 77)(53 204 223 78)(54 205 224 79)(55 206 225 80)(56 207 211 81)(57 208 212 82)(58 209 213 83)(59 210 214 84)(60 196 215 85)
(1 198 189 87)(2 199 190 88)(3 200 191 89)(4 201 192 90)(5 202 193 76)(6 203 194 77)(7 204 195 78)(8 205 181 79)(9 206 182 80)(10 207 183 81)(11 208 184 82)(12 209 185 83)(13 210 186 84)(14 196 187 85)(15 197 188 86)(16 178 111 97)(17 179 112 98)(18 180 113 99)(19 166 114 100)(20 167 115 101)(21 168 116 102)(22 169 117 103)(23 170 118 104)(24 171 119 105)(25 172 120 91)(26 173 106 92)(27 174 107 93)(28 175 108 94)(29 176 109 95)(30 177 110 96)(31 143 125 62)(32 144 126 63)(33 145 127 64)(34 146 128 65)(35 147 129 66)(36 148 130 67)(37 149 131 68)(38 150 132 69)(39 136 133 70)(40 137 134 71)(41 138 135 72)(42 139 121 73)(43 140 122 74)(44 141 123 75)(45 142 124 61)(46 227 216 161)(47 228 217 162)(48 229 218 163)(49 230 219 164)(50 231 220 165)(51 232 221 151)(52 233 222 152)(53 234 223 153)(54 235 224 154)(55 236 225 155)(56 237 211 156)(57 238 212 157)(58 239 213 158)(59 240 214 159)(60 226 215 160)
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), (1,189)(2,190)(3,191)(4,192)(5,193)(6,194)(7,195)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,106)(27,107)(28,108)(29,109)(30,110)(31,221)(32,222)(33,223)(34,224)(35,225)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(43,218)(44,219)(45,220)(46,135)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,166)(88,167)(89,168)(90,169)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,196)(99,197)(100,198)(101,199)(102,200)(103,201)(104,202)(105,203)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,226)(161,227)(162,228)(163,229)(164,230)(165,231), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,16)(14,17)(15,18)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,46)(42,47)(43,48)(44,49)(45,50)(61,165)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,104)(77,105)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,217)(122,218)(123,219)(124,220)(125,221)(126,222)(127,223)(128,224)(129,225)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(166,198)(167,199)(168,200)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206)(175,207)(176,208)(177,209)(178,210)(179,196)(180,197), (1,162,189,228)(2,163,190,229)(3,164,191,230)(4,165,192,231)(5,151,193,232)(6,152,194,233)(7,153,195,234)(8,154,181,235)(9,155,182,236)(10,156,183,237)(11,157,184,238)(12,158,185,239)(13,159,186,240)(14,160,187,226)(15,161,188,227)(16,70,111,136)(17,71,112,137)(18,72,113,138)(19,73,114,139)(20,74,115,140)(21,75,116,141)(22,61,117,142)(23,62,118,143)(24,63,119,144)(25,64,120,145)(26,65,106,146)(27,66,107,147)(28,67,108,148)(29,68,109,149)(30,69,110,150)(31,170,125,104)(32,171,126,105)(33,172,127,91)(34,173,128,92)(35,174,129,93)(36,175,130,94)(37,176,131,95)(38,177,132,96)(39,178,133,97)(40,179,134,98)(41,180,135,99)(42,166,121,100)(43,167,122,101)(44,168,123,102)(45,169,124,103)(46,197,216,86)(47,198,217,87)(48,199,218,88)(49,200,219,89)(50,201,220,90)(51,202,221,76)(52,203,222,77)(53,204,223,78)(54,205,224,79)(55,206,225,80)(56,207,211,81)(57,208,212,82)(58,209,213,83)(59,210,214,84)(60,196,215,85), (1,198,189,87)(2,199,190,88)(3,200,191,89)(4,201,192,90)(5,202,193,76)(6,203,194,77)(7,204,195,78)(8,205,181,79)(9,206,182,80)(10,207,183,81)(11,208,184,82)(12,209,185,83)(13,210,186,84)(14,196,187,85)(15,197,188,86)(16,178,111,97)(17,179,112,98)(18,180,113,99)(19,166,114,100)(20,167,115,101)(21,168,116,102)(22,169,117,103)(23,170,118,104)(24,171,119,105)(25,172,120,91)(26,173,106,92)(27,174,107,93)(28,175,108,94)(29,176,109,95)(30,177,110,96)(31,143,125,62)(32,144,126,63)(33,145,127,64)(34,146,128,65)(35,147,129,66)(36,148,130,67)(37,149,131,68)(38,150,132,69)(39,136,133,70)(40,137,134,71)(41,138,135,72)(42,139,121,73)(43,140,122,74)(44,141,123,75)(45,142,124,61)(46,227,216,161)(47,228,217,162)(48,229,218,163)(49,230,219,164)(50,231,220,165)(51,232,221,151)(52,233,222,152)(53,234,223,153)(54,235,224,154)(55,236,225,155)(56,237,211,156)(57,238,212,157)(58,239,213,158)(59,240,214,159)(60,226,215,160)>;
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), (1,189)(2,190)(3,191)(4,192)(5,193)(6,194)(7,195)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,106)(27,107)(28,108)(29,109)(30,110)(31,221)(32,222)(33,223)(34,224)(35,225)(36,211)(37,212)(38,213)(39,214)(40,215)(41,216)(42,217)(43,218)(44,219)(45,220)(46,135)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,166)(88,167)(89,168)(90,169)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,196)(99,197)(100,198)(101,199)(102,200)(103,201)(104,202)(105,203)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,226)(161,227)(162,228)(163,229)(164,230)(165,231), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,16)(14,17)(15,18)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,46)(42,47)(43,48)(44,49)(45,50)(61,165)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,161)(73,162)(74,163)(75,164)(76,104)(77,105)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,217)(122,218)(123,219)(124,220)(125,221)(126,222)(127,223)(128,224)(129,225)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(166,198)(167,199)(168,200)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206)(175,207)(176,208)(177,209)(178,210)(179,196)(180,197), (1,162,189,228)(2,163,190,229)(3,164,191,230)(4,165,192,231)(5,151,193,232)(6,152,194,233)(7,153,195,234)(8,154,181,235)(9,155,182,236)(10,156,183,237)(11,157,184,238)(12,158,185,239)(13,159,186,240)(14,160,187,226)(15,161,188,227)(16,70,111,136)(17,71,112,137)(18,72,113,138)(19,73,114,139)(20,74,115,140)(21,75,116,141)(22,61,117,142)(23,62,118,143)(24,63,119,144)(25,64,120,145)(26,65,106,146)(27,66,107,147)(28,67,108,148)(29,68,109,149)(30,69,110,150)(31,170,125,104)(32,171,126,105)(33,172,127,91)(34,173,128,92)(35,174,129,93)(36,175,130,94)(37,176,131,95)(38,177,132,96)(39,178,133,97)(40,179,134,98)(41,180,135,99)(42,166,121,100)(43,167,122,101)(44,168,123,102)(45,169,124,103)(46,197,216,86)(47,198,217,87)(48,199,218,88)(49,200,219,89)(50,201,220,90)(51,202,221,76)(52,203,222,77)(53,204,223,78)(54,205,224,79)(55,206,225,80)(56,207,211,81)(57,208,212,82)(58,209,213,83)(59,210,214,84)(60,196,215,85), (1,198,189,87)(2,199,190,88)(3,200,191,89)(4,201,192,90)(5,202,193,76)(6,203,194,77)(7,204,195,78)(8,205,181,79)(9,206,182,80)(10,207,183,81)(11,208,184,82)(12,209,185,83)(13,210,186,84)(14,196,187,85)(15,197,188,86)(16,178,111,97)(17,179,112,98)(18,180,113,99)(19,166,114,100)(20,167,115,101)(21,168,116,102)(22,169,117,103)(23,170,118,104)(24,171,119,105)(25,172,120,91)(26,173,106,92)(27,174,107,93)(28,175,108,94)(29,176,109,95)(30,177,110,96)(31,143,125,62)(32,144,126,63)(33,145,127,64)(34,146,128,65)(35,147,129,66)(36,148,130,67)(37,149,131,68)(38,150,132,69)(39,136,133,70)(40,137,134,71)(41,138,135,72)(42,139,121,73)(43,140,122,74)(44,141,123,75)(45,142,124,61)(46,227,216,161)(47,228,217,162)(48,229,218,163)(49,230,219,164)(50,231,220,165)(51,232,221,151)(52,233,222,152)(53,234,223,153)(54,235,224,154)(55,236,225,155)(56,237,211,156)(57,238,212,157)(58,239,213,158)(59,240,214,159)(60,226,215,160) );
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)], [(1,189),(2,190),(3,191),(4,192),(5,193),(6,194),(7,195),(8,181),(9,182),(10,183),(11,184),(12,185),(13,186),(14,187),(15,188),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,106),(27,107),(28,108),(29,109),(30,110),(31,221),(32,222),(33,223),(34,224),(35,225),(36,211),(37,212),(38,213),(39,214),(40,215),(41,216),(42,217),(43,218),(44,219),(45,220),(46,135),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,177),(84,178),(85,179),(86,180),(87,166),(88,167),(89,168),(90,169),(91,204),(92,205),(93,206),(94,207),(95,208),(96,209),(97,210),(98,196),(99,197),(100,198),(101,199),(102,200),(103,201),(104,202),(105,203),(151,232),(152,233),(153,234),(154,235),(155,236),(156,237),(157,238),(158,239),(159,240),(160,226),(161,227),(162,228),(163,229),(164,230),(165,231)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,16),(14,17),(15,18),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,46),(42,47),(43,48),(44,49),(45,50),(61,165),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,161),(73,162),(74,163),(75,164),(76,104),(77,105),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(106,181),(107,182),(108,183),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(121,217),(122,218),(123,219),(124,220),(125,221),(126,222),(127,223),(128,224),(129,225),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,240),(137,226),(138,227),(139,228),(140,229),(141,230),(142,231),(143,232),(144,233),(145,234),(146,235),(147,236),(148,237),(149,238),(150,239),(166,198),(167,199),(168,200),(169,201),(170,202),(171,203),(172,204),(173,205),(174,206),(175,207),(176,208),(177,209),(178,210),(179,196),(180,197)], [(1,162,189,228),(2,163,190,229),(3,164,191,230),(4,165,192,231),(5,151,193,232),(6,152,194,233),(7,153,195,234),(8,154,181,235),(9,155,182,236),(10,156,183,237),(11,157,184,238),(12,158,185,239),(13,159,186,240),(14,160,187,226),(15,161,188,227),(16,70,111,136),(17,71,112,137),(18,72,113,138),(19,73,114,139),(20,74,115,140),(21,75,116,141),(22,61,117,142),(23,62,118,143),(24,63,119,144),(25,64,120,145),(26,65,106,146),(27,66,107,147),(28,67,108,148),(29,68,109,149),(30,69,110,150),(31,170,125,104),(32,171,126,105),(33,172,127,91),(34,173,128,92),(35,174,129,93),(36,175,130,94),(37,176,131,95),(38,177,132,96),(39,178,133,97),(40,179,134,98),(41,180,135,99),(42,166,121,100),(43,167,122,101),(44,168,123,102),(45,169,124,103),(46,197,216,86),(47,198,217,87),(48,199,218,88),(49,200,219,89),(50,201,220,90),(51,202,221,76),(52,203,222,77),(53,204,223,78),(54,205,224,79),(55,206,225,80),(56,207,211,81),(57,208,212,82),(58,209,213,83),(59,210,214,84),(60,196,215,85)], [(1,198,189,87),(2,199,190,88),(3,200,191,89),(4,201,192,90),(5,202,193,76),(6,203,194,77),(7,204,195,78),(8,205,181,79),(9,206,182,80),(10,207,183,81),(11,208,184,82),(12,209,185,83),(13,210,186,84),(14,196,187,85),(15,197,188,86),(16,178,111,97),(17,179,112,98),(18,180,113,99),(19,166,114,100),(20,167,115,101),(21,168,116,102),(22,169,117,103),(23,170,118,104),(24,171,119,105),(25,172,120,91),(26,173,106,92),(27,174,107,93),(28,175,108,94),(29,176,109,95),(30,177,110,96),(31,143,125,62),(32,144,126,63),(33,145,127,64),(34,146,128,65),(35,147,129,66),(36,148,130,67),(37,149,131,68),(38,150,132,69),(39,136,133,70),(40,137,134,71),(41,138,135,72),(42,139,121,73),(43,140,122,74),(44,141,123,75),(45,142,124,61),(46,227,216,161),(47,228,217,162),(48,229,218,163),(49,230,219,164),(50,231,220,165),(51,232,221,151),(52,233,222,152),(53,234,223,153),(54,235,224,154),(55,236,225,155),(56,237,211,156),(57,238,212,157),(58,239,213,158),(59,240,214,159),(60,226,215,160)]])
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