direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C22.D4, C4⋊C4⋊4C30, C2.7(D4×C30), C22⋊C4⋊4C30, (C22×C4)⋊5C30, (C22×C60)⋊9C2, (C22×C20)⋊9C6, (C2×D4).4C30, C10.70(C6×D4), C6.70(D4×C10), (C22×C12)⋊5C10, (D4×C30).25C2, (C6×D4).11C10, (D4×C10).11C6, (C2×C30).130D4, C30.453(C2×D4), C22.4(D4×C15), C23.12(C2×C30), C30.279(C4○D4), (C2×C60).437C22, (C2×C30).458C23, C22.13(C22×C30), (C22×C30).133C22, (C5×C4⋊C4)⋊13C6, (C15×C4⋊C4)⋊31C2, (C3×C4⋊C4)⋊13C10, (C2×C4).5(C2×C30), C2.6(C15×C4○D4), C6.43(C5×C4○D4), (C2×C6).23(C5×D4), (C5×C22⋊C4)⋊12C6, (C2×C20).67(C2×C6), C10.43(C3×C4○D4), (C2×C10).24(C3×D4), (C3×C22⋊C4)⋊12C10, (C15×C22⋊C4)⋊28C2, (C2×C12).81(C2×C10), (C22×C6).29(C2×C10), (C22×C10).37(C2×C6), (C2×C10).78(C22×C6), (C2×C6).78(C22×C10), SmallGroup(480,928)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C22.D4
G = < a,b,c,d,e | a15=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 232 in 156 conjugacy classes, 88 normal (40 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C30, C22.D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C3×C22.D4, C2×C60, C2×C60, C2×C60, D4×C15, C22×C30, C5×C22.D4, C15×C22⋊C4, C15×C22⋊C4, C15×C4⋊C4, C22×C60, D4×C30, C15×C22.D4
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C4○D4, C2×C10, C3×D4, C22×C6, C30, C22.D4, C5×D4, C22×C10, C6×D4, C3×C4○D4, C2×C30, D4×C10, C5×C4○D4, C3×C22.D4, D4×C15, C22×C30, C5×C22.D4, D4×C30, C15×C4○D4, C15×C22.D4
►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)
(1 178)(2 179)(3 180)(4 166)(5 167)(6 168)(7 169)(8 170)(9 171)(10 172)(11 173)(12 174)(13 175)(14 176)(15 177)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 31)(27 32)(28 33)(29 34)(30 35)(46 193)(47 194)(48 195)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 240)(62 226)(63 227)(64 228)(65 229)(66 230)(67 231)(68 232)(69 233)(70 234)(71 235)(72 236)(73 237)(74 238)(75 239)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(88 165)(89 151)(90 152)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 223)(107 224)(108 225)(109 211)(110 212)(111 213)(112 214)(113 215)(114 216)(115 217)(116 218)(117 219)(118 220)(119 221)(120 222)(136 202)(137 203)(138 204)(139 205)(140 206)(141 207)(142 208)(143 209)(144 210)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)
(1 134)(2 135)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 141)(17 142)(18 143)(19 144)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 136)(27 137)(28 138)(29 139)(30 140)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 164)(47 165)(48 151)(49 152)(50 153)(51 154)(52 155)(53 156)(54 157)(55 158)(56 159)(57 160)(58 161)(59 162)(60 163)(61 216)(62 217)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 225)(71 211)(72 212)(73 213)(74 214)(75 215)(76 182)(77 183)(78 184)(79 185)(80 186)(81 187)(82 188)(83 189)(84 190)(85 191)(86 192)(87 193)(88 194)(89 195)(90 181)(91 171)(92 172)(93 173)(94 174)(95 175)(96 176)(97 177)(98 178)(99 179)(100 180)(101 166)(102 167)(103 168)(104 169)(105 170)(106 232)(107 233)(108 234)(109 235)(110 236)(111 237)(112 238)(113 239)(114 240)(115 226)(116 227)(117 228)(118 229)(119 230)(120 231)
(1 200 98 149)(2 201 99 150)(3 202 100 136)(4 203 101 137)(5 204 102 138)(6 205 103 139)(7 206 104 140)(8 207 105 141)(9 208 91 142)(10 209 92 143)(11 210 93 144)(12 196 94 145)(13 197 95 146)(14 198 96 147)(15 199 97 148)(16 126 36 170)(17 127 37 171)(18 128 38 172)(19 129 39 173)(20 130 40 174)(21 131 41 175)(22 132 42 176)(23 133 43 177)(24 134 44 178)(25 135 45 179)(26 121 31 180)(27 122 32 166)(28 123 33 167)(29 124 34 168)(30 125 35 169)(46 215 87 113)(47 216 88 114)(48 217 89 115)(49 218 90 116)(50 219 76 117)(51 220 77 118)(52 221 78 119)(53 222 79 120)(54 223 80 106)(55 224 81 107)(56 225 82 108)(57 211 83 109)(58 212 84 110)(59 213 85 111)(60 214 86 112)(61 194 240 165)(62 195 226 151)(63 181 227 152)(64 182 228 153)(65 183 229 154)(66 184 230 155)(67 185 231 156)(68 186 232 157)(69 187 233 158)(70 188 234 159)(71 189 235 160)(72 190 236 161)(73 191 237 162)(74 192 238 163)(75 193 239 164)
(1 239)(2 240)(3 226)(4 227)(5 228)(6 229)(7 230)(8 231)(9 232)(10 233)(11 234)(12 235)(13 236)(14 237)(15 238)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 151)(27 152)(28 153)(29 154)(30 155)(31 195)(32 181)(33 182)(34 183)(35 184)(36 185)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 193)(45 194)(46 149)(47 150)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 204)(77 205)(78 206)(79 207)(80 208)(81 209)(82 210)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(166 218)(167 219)(168 220)(169 221)(170 222)(171 223)(172 224)(173 225)(174 211)(175 212)(176 213)(177 214)(178 215)(179 216)(180 217)
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), (1,178)(2,179)(3,180)(4,166)(5,167)(6,168)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,31)(27,32)(28,33)(29,34)(30,35)(46,193)(47,194)(48,195)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,240)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165)(89,151)(90,152)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,223)(107,224)(108,225)(109,211)(110,212)(111,213)(112,214)(113,215)(114,216)(115,217)(116,218)(117,219)(118,220)(119,221)(120,222)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201), (1,134)(2,135)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,136)(27,137)(28,138)(29,139)(30,140)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,164)(47,165)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,216)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,211)(72,212)(73,213)(74,214)(75,215)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(85,191)(86,192)(87,193)(88,194)(89,195)(90,181)(91,171)(92,172)(93,173)(94,174)(95,175)(96,176)(97,177)(98,178)(99,179)(100,180)(101,166)(102,167)(103,168)(104,169)(105,170)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,226)(116,227)(117,228)(118,229)(119,230)(120,231), (1,200,98,149)(2,201,99,150)(3,202,100,136)(4,203,101,137)(5,204,102,138)(6,205,103,139)(7,206,104,140)(8,207,105,141)(9,208,91,142)(10,209,92,143)(11,210,93,144)(12,196,94,145)(13,197,95,146)(14,198,96,147)(15,199,97,148)(16,126,36,170)(17,127,37,171)(18,128,38,172)(19,129,39,173)(20,130,40,174)(21,131,41,175)(22,132,42,176)(23,133,43,177)(24,134,44,178)(25,135,45,179)(26,121,31,180)(27,122,32,166)(28,123,33,167)(29,124,34,168)(30,125,35,169)(46,215,87,113)(47,216,88,114)(48,217,89,115)(49,218,90,116)(50,219,76,117)(51,220,77,118)(52,221,78,119)(53,222,79,120)(54,223,80,106)(55,224,81,107)(56,225,82,108)(57,211,83,109)(58,212,84,110)(59,213,85,111)(60,214,86,112)(61,194,240,165)(62,195,226,151)(63,181,227,152)(64,182,228,153)(65,183,229,154)(66,184,230,155)(67,185,231,156)(68,186,232,157)(69,187,233,158)(70,188,234,159)(71,189,235,160)(72,190,236,161)(73,191,237,162)(74,192,238,163)(75,193,239,164), (1,239)(2,240)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,151)(27,152)(28,153)(29,154)(30,155)(31,195)(32,181)(33,182)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,149)(47,150)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,211)(175,212)(176,213)(177,214)(178,215)(179,216)(180,217)>;
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), (1,178)(2,179)(3,180)(4,166)(5,167)(6,168)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,31)(27,32)(28,33)(29,34)(30,35)(46,193)(47,194)(48,195)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,240)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165)(89,151)(90,152)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,223)(107,224)(108,225)(109,211)(110,212)(111,213)(112,214)(113,215)(114,216)(115,217)(116,218)(117,219)(118,220)(119,221)(120,222)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201), (1,134)(2,135)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,136)(27,137)(28,138)(29,139)(30,140)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,164)(47,165)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,216)(62,217)(63,218)(64,219)(65,220)(66,221)(67,222)(68,223)(69,224)(70,225)(71,211)(72,212)(73,213)(74,214)(75,215)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(85,191)(86,192)(87,193)(88,194)(89,195)(90,181)(91,171)(92,172)(93,173)(94,174)(95,175)(96,176)(97,177)(98,178)(99,179)(100,180)(101,166)(102,167)(103,168)(104,169)(105,170)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,226)(116,227)(117,228)(118,229)(119,230)(120,231), (1,200,98,149)(2,201,99,150)(3,202,100,136)(4,203,101,137)(5,204,102,138)(6,205,103,139)(7,206,104,140)(8,207,105,141)(9,208,91,142)(10,209,92,143)(11,210,93,144)(12,196,94,145)(13,197,95,146)(14,198,96,147)(15,199,97,148)(16,126,36,170)(17,127,37,171)(18,128,38,172)(19,129,39,173)(20,130,40,174)(21,131,41,175)(22,132,42,176)(23,133,43,177)(24,134,44,178)(25,135,45,179)(26,121,31,180)(27,122,32,166)(28,123,33,167)(29,124,34,168)(30,125,35,169)(46,215,87,113)(47,216,88,114)(48,217,89,115)(49,218,90,116)(50,219,76,117)(51,220,77,118)(52,221,78,119)(53,222,79,120)(54,223,80,106)(55,224,81,107)(56,225,82,108)(57,211,83,109)(58,212,84,110)(59,213,85,111)(60,214,86,112)(61,194,240,165)(62,195,226,151)(63,181,227,152)(64,182,228,153)(65,183,229,154)(66,184,230,155)(67,185,231,156)(68,186,232,157)(69,187,233,158)(70,188,234,159)(71,189,235,160)(72,190,236,161)(73,191,237,162)(74,192,238,163)(75,193,239,164), (1,239)(2,240)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,151)(27,152)(28,153)(29,154)(30,155)(31,195)(32,181)(33,182)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,149)(47,150)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,211)(175,212)(176,213)(177,214)(178,215)(179,216)(180,217) );
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)], [(1,178),(2,179),(3,180),(4,166),(5,167),(6,168),(7,169),(8,170),(9,171),(10,172),(11,173),(12,174),(13,175),(14,176),(15,177),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,31),(27,32),(28,33),(29,34),(30,35),(46,193),(47,194),(48,195),(49,181),(50,182),(51,183),(52,184),(53,185),(54,186),(55,187),(56,188),(57,189),(58,190),(59,191),(60,192),(61,240),(62,226),(63,227),(64,228),(65,229),(66,230),(67,231),(68,232),(69,233),(70,234),(71,235),(72,236),(73,237),(74,238),(75,239),(76,153),(77,154),(78,155),(79,156),(80,157),(81,158),(82,159),(83,160),(84,161),(85,162),(86,163),(87,164),(88,165),(89,151),(90,152),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,223),(107,224),(108,225),(109,211),(110,212),(111,213),(112,214),(113,215),(114,216),(115,217),(116,218),(117,219),(118,220),(119,221),(120,222),(136,202),(137,203),(138,204),(139,205),(140,206),(141,207),(142,208),(143,209),(144,210),(145,196),(146,197),(147,198),(148,199),(149,200),(150,201)], [(1,134),(2,135),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,141),(17,142),(18,143),(19,144),(20,145),(21,146),(22,147),(23,148),(24,149),(25,150),(26,136),(27,137),(28,138),(29,139),(30,140),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,196),(41,197),(42,198),(43,199),(44,200),(45,201),(46,164),(47,165),(48,151),(49,152),(50,153),(51,154),(52,155),(53,156),(54,157),(55,158),(56,159),(57,160),(58,161),(59,162),(60,163),(61,216),(62,217),(63,218),(64,219),(65,220),(66,221),(67,222),(68,223),(69,224),(70,225),(71,211),(72,212),(73,213),(74,214),(75,215),(76,182),(77,183),(78,184),(79,185),(80,186),(81,187),(82,188),(83,189),(84,190),(85,191),(86,192),(87,193),(88,194),(89,195),(90,181),(91,171),(92,172),(93,173),(94,174),(95,175),(96,176),(97,177),(98,178),(99,179),(100,180),(101,166),(102,167),(103,168),(104,169),(105,170),(106,232),(107,233),(108,234),(109,235),(110,236),(111,237),(112,238),(113,239),(114,240),(115,226),(116,227),(117,228),(118,229),(119,230),(120,231)], [(1,200,98,149),(2,201,99,150),(3,202,100,136),(4,203,101,137),(5,204,102,138),(6,205,103,139),(7,206,104,140),(8,207,105,141),(9,208,91,142),(10,209,92,143),(11,210,93,144),(12,196,94,145),(13,197,95,146),(14,198,96,147),(15,199,97,148),(16,126,36,170),(17,127,37,171),(18,128,38,172),(19,129,39,173),(20,130,40,174),(21,131,41,175),(22,132,42,176),(23,133,43,177),(24,134,44,178),(25,135,45,179),(26,121,31,180),(27,122,32,166),(28,123,33,167),(29,124,34,168),(30,125,35,169),(46,215,87,113),(47,216,88,114),(48,217,89,115),(49,218,90,116),(50,219,76,117),(51,220,77,118),(52,221,78,119),(53,222,79,120),(54,223,80,106),(55,224,81,107),(56,225,82,108),(57,211,83,109),(58,212,84,110),(59,213,85,111),(60,214,86,112),(61,194,240,165),(62,195,226,151),(63,181,227,152),(64,182,228,153),(65,183,229,154),(66,184,230,155),(67,185,231,156),(68,186,232,157),(69,187,233,158),(70,188,234,159),(71,189,235,160),(72,190,236,161),(73,191,237,162),(74,192,238,163),(75,193,239,164)], [(1,239),(2,240),(3,226),(4,227),(5,228),(6,229),(7,230),(8,231),(9,232),(10,233),(11,234),(12,235),(13,236),(14,237),(15,238),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,165),(26,151),(27,152),(28,153),(29,154),(30,155),(31,195),(32,181),(33,182),(34,183),(35,184),(36,185),(37,186),(38,187),(39,188),(40,189),(41,190),(42,191),(43,192),(44,193),(45,194),(46,149),(47,150),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,97),(75,98),(76,204),(77,205),(78,206),(79,207),(80,208),(81,209),(82,210),(83,196),(84,197),(85,198),(86,199),(87,200),(88,201),(89,202),(90,203),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126),(166,218),(167,219),(168,220),(169,221),(170,222),(171,223),(172,224),(173,225),(174,211),(175,212),(176,213),(177,214),(178,215),(179,216),(180,217)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 12A | ··· | 12H | 12I | ··· | 12N | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20AB | 30A | ··· | 30X | 30Y | ··· | 30AN | 30AO | ··· | 30AV | 60A | ··· | 60AF | 60AG | ··· | 60BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 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 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | D4 | C4○D4 | C3×D4 | C5×D4 | C3×C4○D4 | C5×C4○D4 | D4×C15 | C15×C4○D4 |
| kernel | C15×C22.D4 | C15×C22⋊C4 | C15×C4⋊C4 | C22×C60 | D4×C30 | C5×C22.D4 | C3×C22.D4 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C30 | C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 2 | 4 | 6 | 4 | 2 | 2 | 12 | 8 | 4 | 4 | 8 | 24 | 16 | 8 | 8 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×C22.D4 ►in GL4(𝔽61) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 0 | 50 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 17 | 9 |
| 0 | 0 | 22 | 44 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 48 | 4 |
| 0 | 0 | 19 | 13 |
G:=sub<GL(4,GF(61))| [42,0,0,0,0,42,0,0,0,0,25,0,0,0,0,25],[60,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[0,11,0,0,50,0,0,0,0,0,17,22,0,0,9,44],[0,1,0,0,1,0,0,0,0,0,48,19,0,0,4,13] >;
C15×C22.D4 in GAP, Magma, Sage, TeX
C_{15}\times C_2^2.D_4 % in TeX
G:=Group("C15xC2^2.D4"); // GroupNames label
G:=SmallGroup(480,928);
// by ID
G=gap.SmallGroup(480,928);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,5126,646]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations