direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊2D15, D4⋊5D30, C30.59C24, C60.83C23, C23.24D30, D30.25C23, Dic30⋊22C22, Dic15.45C23, (C6×D4)⋊6D5, (D4×C10)⋊6S3, (D4×C30)⋊6C2, (C5×D4)⋊21D6, (C2×D4)⋊8D15, (C3×D4)⋊21D10, (C2×C4).60D30, C30⋊14(C4○D4), C6⋊5(D4⋊2D5), (C2×C20).170D6, C15⋊7D4⋊9C22, (C2×C30).8C23, C10⋊5(D4⋊2S3), C6.59(C23×D5), C2.7(C23×D15), (C2×Dic30)⋊15C2, (C2×C12).168D10, (C4×D15)⋊17C22, (D4×C15)⋊23C22, (C2×C60).86C22, C10.59(S3×C23), (C22×C10).85D6, C4.20(C22×D15), (C22×C6).70D10, C20.133(C22×S3), (C22×Dic15)⋊8C2, C12.131(C22×D5), C22.1(C22×D15), (C2×Dic15)⋊26C22, (C22×C30).24C22, (C22×D15).91C22, (C2×C4×D15)⋊4C2, C15⋊23(C2×C4○D4), C3⋊6(C2×D4⋊2D5), C5⋊6(C2×D4⋊2S3), (C2×C15⋊7D4)⋊10C2, (C2×C6).15(C22×D5), (C2×C10).317(C22×S3), SmallGroup(480,1170)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1556 in 328 conjugacy classes, 127 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×4], Dic3 [×6], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C2×C10 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×11], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], D15 [×2], C30, C30 [×2], C30 [×4], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5 [×11], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×Dic6, S3×C2×C4, D4⋊2S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, Dic15 [×6], C60 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×Dic10, C2×C4×D5, D4⋊2D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×D4⋊2S3, Dic30 [×4], C4×D15 [×4], C2×Dic15, C2×Dic15 [×10], C15⋊7D4 [×8], C2×C60, D4×C15 [×4], C22×D15, C22×C30 [×2], C2×D4⋊2D5, C2×Dic30, C2×C4×D15, D4⋊2D15 [×8], C22×Dic15 [×2], C2×C15⋊7D4 [×2], D4×C30, C2×D4⋊2D15
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], D15, C2×C4○D4, C22×D5 [×7], D4⋊2S3 [×2], S3×C23, D30 [×7], D4⋊2D5 [×2], C23×D5, C2×D4⋊2S3, C22×D15 [×7], C2×D4⋊2D5, D4⋊2D15 [×2], C23×D15, C2×D4⋊2D15
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
►On 240 points
Generators in S240
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 42)(17 43)(18 44)(19 45)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 106)(72 107)(73 108)(74 109)(75 110)(76 103)(77 104)(78 105)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(121 167)(122 168)(123 169)(124 170)(125 171)(126 172)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 166)(136 155)(137 156)(138 157)(139 158)(140 159)(141 160)(142 161)(143 162)(144 163)(145 164)(146 165)(147 151)(148 152)(149 153)(150 154)(181 230)(182 231)(183 232)(184 233)(185 234)(186 235)(187 236)(188 237)(189 238)(190 239)(191 240)(192 226)(193 227)(194 228)(195 229)(196 211)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 218)(204 219)(205 220)(206 221)(207 222)(208 223)(209 224)(210 225)
(1 84 22 61)(2 85 23 62)(3 86 24 63)(4 87 25 64)(5 88 26 65)(6 89 27 66)(7 90 28 67)(8 76 29 68)(9 77 30 69)(10 78 16 70)(11 79 17 71)(12 80 18 72)(13 81 19 73)(14 82 20 74)(15 83 21 75)(31 109 53 94)(32 110 54 95)(33 111 55 96)(34 112 56 97)(35 113 57 98)(36 114 58 99)(37 115 59 100)(38 116 60 101)(39 117 46 102)(40 118 47 103)(41 119 48 104)(42 120 49 105)(43 106 50 91)(44 107 51 92)(45 108 52 93)(121 200 140 185)(122 201 141 186)(123 202 142 187)(124 203 143 188)(125 204 144 189)(126 205 145 190)(127 206 146 191)(128 207 147 192)(129 208 148 193)(130 209 149 194)(131 210 150 195)(132 196 136 181)(133 197 137 182)(134 198 138 183)(135 199 139 184)(151 226 174 222)(152 227 175 223)(153 228 176 224)(154 229 177 225)(155 230 178 211)(156 231 179 212)(157 232 180 213)(158 233 166 214)(159 234 167 215)(160 235 168 216)(161 236 169 217)(162 237 170 218)(163 238 171 219)(164 239 172 220)(165 240 173 221)
(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 76)(69 77)(70 78)(71 79)(72 80)(73 81)(74 82)(75 83)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 136)(133 137)(134 138)(135 139)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)(157 180)(158 166)(159 167)(160 168)(161 169)(162 170)(163 171)(164 172)(165 173)
(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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 145)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 136)(26 150)(27 149)(28 148)(29 147)(30 146)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 165)(42 164)(43 163)(44 162)(45 161)(46 175)(47 174)(48 173)(49 172)(50 171)(51 170)(52 169)(53 168)(54 167)(55 166)(56 180)(57 179)(58 178)(59 177)(60 176)(61 184)(62 183)(63 182)(64 181)(65 195)(66 194)(67 193)(68 192)(69 191)(70 190)(71 189)(72 188)(73 187)(74 186)(75 185)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 210)(89 209)(90 208)(91 219)(92 218)(93 217)(94 216)(95 215)(96 214)(97 213)(98 212)(99 211)(100 225)(101 224)(102 223)(103 222)(104 221)(105 220)(106 238)(107 237)(108 236)(109 235)(110 234)(111 233)(112 232)(113 231)(114 230)(115 229)(116 228)(117 227)(118 226)(119 240)(120 239)
G:=sub<Sym(240)| (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,42)(17,43)(18,44)(19,45)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,106)(72,107)(73,108)(74,109)(75,110)(76,103)(77,104)(78,105)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,166)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)(144,163)(145,164)(146,165)(147,151)(148,152)(149,153)(150,154)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,226)(193,227)(194,228)(195,229)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (1,84,22,61)(2,85,23,62)(3,86,24,63)(4,87,25,64)(5,88,26,65)(6,89,27,66)(7,90,28,67)(8,76,29,68)(9,77,30,69)(10,78,16,70)(11,79,17,71)(12,80,18,72)(13,81,19,73)(14,82,20,74)(15,83,21,75)(31,109,53,94)(32,110,54,95)(33,111,55,96)(34,112,56,97)(35,113,57,98)(36,114,58,99)(37,115,59,100)(38,116,60,101)(39,117,46,102)(40,118,47,103)(41,119,48,104)(42,120,49,105)(43,106,50,91)(44,107,51,92)(45,108,52,93)(121,200,140,185)(122,201,141,186)(123,202,142,187)(124,203,143,188)(125,204,144,189)(126,205,145,190)(127,206,146,191)(128,207,147,192)(129,208,148,193)(130,209,149,194)(131,210,150,195)(132,196,136,181)(133,197,137,182)(134,198,138,183)(135,199,139,184)(151,226,174,222)(152,227,175,223)(153,228,176,224)(154,229,177,225)(155,230,178,211)(156,231,179,212)(157,232,180,213)(158,233,166,214)(159,234,167,215)(160,235,168,216)(161,236,169,217)(162,237,170,218)(163,238,171,219)(164,239,172,220)(165,240,173,221), (61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,76)(69,77)(70,78)(71,79)(72,80)(73,81)(74,82)(75,83)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,136)(133,137)(134,138)(135,139)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,166)(159,167)(160,168)(161,169)(162,170)(163,171)(164,172)(165,173), (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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,145)(17,144)(18,143)(19,142)(20,141)(21,140)(22,139)(23,138)(24,137)(25,136)(26,150)(27,149)(28,148)(29,147)(30,146)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,165)(42,164)(43,163)(44,162)(45,161)(46,175)(47,174)(48,173)(49,172)(50,171)(51,170)(52,169)(53,168)(54,167)(55,166)(56,180)(57,179)(58,178)(59,177)(60,176)(61,184)(62,183)(63,182)(64,181)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,189)(72,188)(73,187)(74,186)(75,185)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,210)(89,209)(90,208)(91,219)(92,218)(93,217)(94,216)(95,215)(96,214)(97,213)(98,212)(99,211)(100,225)(101,224)(102,223)(103,222)(104,221)(105,220)(106,238)(107,237)(108,236)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,228)(117,227)(118,226)(119,240)(120,239)>;
G:=Group( (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,42)(17,43)(18,44)(19,45)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,106)(72,107)(73,108)(74,109)(75,110)(76,103)(77,104)(78,105)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,166)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)(144,163)(145,164)(146,165)(147,151)(148,152)(149,153)(150,154)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,226)(193,227)(194,228)(195,229)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (1,84,22,61)(2,85,23,62)(3,86,24,63)(4,87,25,64)(5,88,26,65)(6,89,27,66)(7,90,28,67)(8,76,29,68)(9,77,30,69)(10,78,16,70)(11,79,17,71)(12,80,18,72)(13,81,19,73)(14,82,20,74)(15,83,21,75)(31,109,53,94)(32,110,54,95)(33,111,55,96)(34,112,56,97)(35,113,57,98)(36,114,58,99)(37,115,59,100)(38,116,60,101)(39,117,46,102)(40,118,47,103)(41,119,48,104)(42,120,49,105)(43,106,50,91)(44,107,51,92)(45,108,52,93)(121,200,140,185)(122,201,141,186)(123,202,142,187)(124,203,143,188)(125,204,144,189)(126,205,145,190)(127,206,146,191)(128,207,147,192)(129,208,148,193)(130,209,149,194)(131,210,150,195)(132,196,136,181)(133,197,137,182)(134,198,138,183)(135,199,139,184)(151,226,174,222)(152,227,175,223)(153,228,176,224)(154,229,177,225)(155,230,178,211)(156,231,179,212)(157,232,180,213)(158,233,166,214)(159,234,167,215)(160,235,168,216)(161,236,169,217)(162,237,170,218)(163,238,171,219)(164,239,172,220)(165,240,173,221), (61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,76)(69,77)(70,78)(71,79)(72,80)(73,81)(74,82)(75,83)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,136)(133,137)(134,138)(135,139)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,166)(159,167)(160,168)(161,169)(162,170)(163,171)(164,172)(165,173), (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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,145)(17,144)(18,143)(19,142)(20,141)(21,140)(22,139)(23,138)(24,137)(25,136)(26,150)(27,149)(28,148)(29,147)(30,146)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,165)(42,164)(43,163)(44,162)(45,161)(46,175)(47,174)(48,173)(49,172)(50,171)(51,170)(52,169)(53,168)(54,167)(55,166)(56,180)(57,179)(58,178)(59,177)(60,176)(61,184)(62,183)(63,182)(64,181)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,189)(72,188)(73,187)(74,186)(75,185)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,210)(89,209)(90,208)(91,219)(92,218)(93,217)(94,216)(95,215)(96,214)(97,213)(98,212)(99,211)(100,225)(101,224)(102,223)(103,222)(104,221)(105,220)(106,238)(107,237)(108,236)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,228)(117,227)(118,226)(119,240)(120,239) );
G=PermutationGroup([(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,42),(17,43),(18,44),(19,45),(20,31),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,106),(72,107),(73,108),(74,109),(75,110),(76,103),(77,104),(78,105),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(121,167),(122,168),(123,169),(124,170),(125,171),(126,172),(127,173),(128,174),(129,175),(130,176),(131,177),(132,178),(133,179),(134,180),(135,166),(136,155),(137,156),(138,157),(139,158),(140,159),(141,160),(142,161),(143,162),(144,163),(145,164),(146,165),(147,151),(148,152),(149,153),(150,154),(181,230),(182,231),(183,232),(184,233),(185,234),(186,235),(187,236),(188,237),(189,238),(190,239),(191,240),(192,226),(193,227),(194,228),(195,229),(196,211),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,218),(204,219),(205,220),(206,221),(207,222),(208,223),(209,224),(210,225)], [(1,84,22,61),(2,85,23,62),(3,86,24,63),(4,87,25,64),(5,88,26,65),(6,89,27,66),(7,90,28,67),(8,76,29,68),(9,77,30,69),(10,78,16,70),(11,79,17,71),(12,80,18,72),(13,81,19,73),(14,82,20,74),(15,83,21,75),(31,109,53,94),(32,110,54,95),(33,111,55,96),(34,112,56,97),(35,113,57,98),(36,114,58,99),(37,115,59,100),(38,116,60,101),(39,117,46,102),(40,118,47,103),(41,119,48,104),(42,120,49,105),(43,106,50,91),(44,107,51,92),(45,108,52,93),(121,200,140,185),(122,201,141,186),(123,202,142,187),(124,203,143,188),(125,204,144,189),(126,205,145,190),(127,206,146,191),(128,207,147,192),(129,208,148,193),(130,209,149,194),(131,210,150,195),(132,196,136,181),(133,197,137,182),(134,198,138,183),(135,199,139,184),(151,226,174,222),(152,227,175,223),(153,228,176,224),(154,229,177,225),(155,230,178,211),(156,231,179,212),(157,232,180,213),(158,233,166,214),(159,234,167,215),(160,235,168,216),(161,236,169,217),(162,237,170,218),(163,238,171,219),(164,239,172,220),(165,240,173,221)], [(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,76),(69,77),(70,78),(71,79),(72,80),(73,81),(74,82),(75,83),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,136),(133,137),(134,138),(135,139),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179),(157,180),(158,166),(159,167),(160,168),(161,169),(162,170),(163,171),(164,172),(165,173)], [(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)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,145),(17,144),(18,143),(19,142),(20,141),(21,140),(22,139),(23,138),(24,137),(25,136),(26,150),(27,149),(28,148),(29,147),(30,146),(31,160),(32,159),(33,158),(34,157),(35,156),(36,155),(37,154),(38,153),(39,152),(40,151),(41,165),(42,164),(43,163),(44,162),(45,161),(46,175),(47,174),(48,173),(49,172),(50,171),(51,170),(52,169),(53,168),(54,167),(55,166),(56,180),(57,179),(58,178),(59,177),(60,176),(61,184),(62,183),(63,182),(64,181),(65,195),(66,194),(67,193),(68,192),(69,191),(70,190),(71,189),(72,188),(73,187),(74,186),(75,185),(76,207),(77,206),(78,205),(79,204),(80,203),(81,202),(82,201),(83,200),(84,199),(85,198),(86,197),(87,196),(88,210),(89,209),(90,208),(91,219),(92,218),(93,217),(94,216),(95,215),(96,214),(97,213),(98,212),(99,211),(100,225),(101,224),(102,223),(103,222),(104,221),(105,220),(106,238),(107,237),(108,236),(109,235),(110,234),(111,233),(112,232),(113,231),(114,230),(115,229),(116,228),(117,227),(118,226),(119,240),(120,239)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 21 | 0 | 0 |
| 0 | 58 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 53 | 38 |
| 0 | 0 | 0 | 23 | 5 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 50 | 13 | 0 | 0 |
| 0 | 33 | 11 | 0 | 0 |
| 0 | 0 | 0 | 6 | 47 |
| 0 | 0 | 0 | 33 | 55 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,58,0,0,0,21,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,21,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,23,0,0,0,38,5],[60,0,0,0,0,0,50,33,0,0,0,13,11,0,0,0,0,0,6,33,0,0,0,47,55] >;
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | D4⋊2S3 | D4⋊2D5 | D4⋊2D15 |
| kernel | C2×D4⋊2D15 | C2×Dic30 | C2×C4×D15 | D4⋊2D15 | C22×Dic15 | C2×C15⋊7D4 | D4×C30 | D4×C10 | C6×D4 | C2×C20 | C5×D4 | C22×C10 | C30 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C10 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 4 | 4 | 4 | 16 | 8 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_2D_{15} % in TeX
G:=Group("C2xD4:2D15"); // GroupNames label
G:=SmallGroup(480,1170);
// by ID
G=gap.SmallGroup(480,1170);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,185,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=e^2=1,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,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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