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G = C2×D42D15order 480 = 25·3·5

Direct product of C2 and D42D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D42D15, D45D30, C60.83C23, C30.59C24, C23.24D30, D30.25C23, Dic3022C22, Dic15.45C23, (C6×D4)⋊6D5, (C5×D4)⋊21D6, (C2×D4)⋊8D15, (D4×C30)⋊6C2, (D4×C10)⋊6S3, (C3×D4)⋊21D10, (C2×C4).60D30, C3014(C4○D4), C65(D42D5), (C2×C20).170D6, C157D49C22, (C2×C30).8C23, C105(D42S3), 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×C6).70D10, C4.20(C22×D15), (C22×C10).85D6, 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, C1523(C2×C4○D4), C36(C2×D42D5), C56(C2×D42S3), (C2×C157D4)⋊10C2, (C2×C6).15(C22×D5), (C2×C10).317(C22×S3), SmallGroup(480,1170)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D42D15
C1C5C15C30D30C22×D15C2×C4×D15 — C2×D42D15
C15C30 — C2×D42D15
C1C22C2×D4

Generators and relations for C2×D42D15
 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 >

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, D42S3 [×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, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×D42S3, Dic30 [×4], C4×D15 [×4], C2×Dic15, C2×Dic15 [×10], C157D4 [×8], C2×C60, D4×C15 [×4], C22×D15, C22×C30 [×2], C2×D42D5, C2×Dic30, C2×C4×D15, D42D15 [×8], C22×Dic15 [×2], C2×C157D4 [×2], D4×C30, C2×D42D15
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], D42S3 [×2], S3×C23, D30 [×7], D42D5 [×2], C23×D5, C2×D42S3, C22×D15 [×7], C2×D42D5, D42D15 [×2], C23×D15, C2×D42D15

Smallest permutation representation of C2×D42D15
On 240 points
Generators in S240
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 106)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 166)(134 167)(135 168)(136 154)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 161)(144 162)(145 163)(146 164)(147 165)(148 151)(149 152)(150 153)(181 227)(182 228)(183 229)(184 230)(185 231)(186 232)(187 233)(188 234)(189 235)(190 236)(191 237)(192 238)(193 239)(194 240)(195 226)(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 78 19 65)(2 79 20 66)(3 80 21 67)(4 81 22 68)(5 82 23 69)(6 83 24 70)(7 84 25 71)(8 85 26 72)(9 86 27 73)(10 87 28 74)(11 88 29 75)(12 89 30 61)(13 90 16 62)(14 76 17 63)(15 77 18 64)(31 108 49 105)(32 109 50 91)(33 110 51 92)(34 111 52 93)(35 112 53 94)(36 113 54 95)(37 114 55 96)(38 115 56 97)(39 116 57 98)(40 117 58 99)(41 118 59 100)(42 119 60 101)(43 120 46 102)(44 106 47 103)(45 107 48 104)(121 203 140 184)(122 204 141 185)(123 205 142 186)(124 206 143 187)(125 207 144 188)(126 208 145 189)(127 209 146 190)(128 210 147 191)(129 196 148 192)(130 197 149 193)(131 198 150 194)(132 199 136 195)(133 200 137 181)(134 201 138 182)(135 202 139 183)(151 238 177 211)(152 239 178 212)(153 240 179 213)(154 226 180 214)(155 227 166 215)(156 228 167 216)(157 229 168 217)(158 230 169 218)(159 231 170 219)(160 232 171 220)(161 233 172 221)(162 234 173 222)(163 235 174 223)(164 236 175 224)(165 237 176 225)
(61 89)(62 90)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 106)(104 107)(105 108)(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)(151 177)(152 178)(153 179)(154 180)(155 166)(156 167)(157 168)(158 169)(159 170)(160 171)(161 172)(162 173)(163 174)(164 175)(165 176)
(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)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 135)(14 134)(15 133)(16 139)(17 138)(18 137)(19 136)(20 150)(21 149)(22 148)(23 147)(24 146)(25 145)(26 144)(27 143)(28 142)(29 141)(30 140)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 165)(39 164)(40 163)(41 162)(42 161)(43 160)(44 159)(45 158)(46 171)(47 170)(48 169)(49 168)(50 167)(51 166)(52 180)(53 179)(54 178)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(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)(76 201)(77 200)(78 199)(79 198)(80 197)(81 196)(82 210)(83 209)(84 208)(85 207)(86 206)(87 205)(88 204)(89 203)(90 202)(91 216)(92 215)(93 214)(94 213)(95 212)(96 211)(97 225)(98 224)(99 223)(100 222)(101 221)(102 220)(103 219)(104 218)(105 217)(106 231)(107 230)(108 229)(109 228)(110 227)(111 226)(112 240)(113 239)(114 238)(115 237)(116 236)(117 235)(118 234)(119 233)(120 232)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222223444444444455666666610···1010···101212151515152020202030···3030···3060···60
size11112222303022215151515303030302222244442···24···444222244442···24···44···4

90 irreducible representations

dim11111112222222222222444
type+++++++++++++++++++---
imageC1C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10D10D15D30D30D30D42S3D42D5D42D15
kernelC2×D42D15C2×Dic30C2×C4×D15D42D15C22×Dic15C2×C157D4D4×C30D4×C10C6×D4C2×C20C5×D4C22×C10C30C2×C12C3×D4C22×C6C2×D4C2×C4D4C23C10C6C2
# reps111822112142428444168248

Matrix representation of C2×D42D15 in GL5(𝔽61)

600000
060000
006000
00010
00001
,
600000
012100
0586000
00010
00001
,
600000
012100
006000
00010
00001
,
10000
01000
00100
0005338
000235
,
600000
0501300
0331100
000647
0003355

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] >;

C2×D42D15 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

׿
×
𝔽