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G = C2×Dic3.F5  order 480 = 25·3·5

Direct product of C2 and Dic3.F5

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

Aliases: C2×Dic3.F5, C30⋊2M4(2), C5⋊C8⋊7D6, C6⋊1(C4.F5), C10⋊1(C8⋊S3), C15⋊7(C2×M4(2)), D30.15(C2×C4), D30.C2.6C4, (C2×Dic3).7F5, C15⋊C8⋊10C22, C22.23(S3×F5), C6.28(C22×F5), C30.28(C22×C4), Dic5.32(C4×S3), Dic3.10(C2×F5), (C22×D15).7C4, (C2×Dic5).152D6, (C10×Dic3).10C4, D30.C2.18C22, Dic5.40(C22×S3), (C3×Dic5).38C23, (C6×Dic5).149C22, (C6×C5⋊C8)⋊7C2, (C2×C5⋊C8)⋊5S3, C5⋊2(C2×C8⋊S3), C3⋊2(C2×C4.F5), C2.28(C2×S3×F5), C10.28(S3×C2×C4), (C2×C15⋊C8)⋊7C2, (C3×C5⋊C8)⋊10C22, (C2×C6).25(C2×F5), (C2×C10).23(C4×S3), (C2×C30).23(C2×C4), (C2×D30.C2).13C2, (C3×Dic5).30(C2×C4), (C5×Dic3).16(C2×C4), SmallGroup(480,1009)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C2×Dic3.F5
Chief series
C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — Dic3.F5 — C2×Dic3.F5
Lower central
C15 — C30 — C2×Dic3.F5

Subgroups: 692 in 136 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×4], C2×C4 [×6], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], D30 [×2], D30 [×2], C2×C30, C4.F5 [×4], C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C8⋊S3, C3×C5⋊C8 [×2], C15⋊C8 [×2], D30.C2 [×4], C6×Dic5, C10×Dic3, C22×D15, C2×C4.F5, Dic3.F5 [×4], C6×C5⋊C8, C2×C15⋊C8, C2×D30.C2, C2×Dic3.F5

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, F5, C4×S3 [×2], C22×S3, C2×M4(2), C2×F5 [×3], C8⋊S3 [×2], S3×C2×C4, C4.F5 [×2], C22×F5, C2×C8⋊S3, S3×F5, C2×C4.F5, Dic3.F5 [×2], C2×S3×F5, C2×Dic3.F5

Generators and relations
 G = < a,b,c,d,e | a2=b6=d5=1, c2=e4=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽241)

2400000000
0240000000
0024000000
0002400000
00001000
00000100
00000010
00000001
,
2400000000
0240000000
002402400000
00100000
00001000
00000100
00000010
00000001
,
640000000
0177000000
00100000
002402400000
0000240000
0000024000
0000002400
0000000240
,
10000000
01000000
00100000
00010000
0000000240
0000100240
0000010240
0000001240
,
01000000
1770000000
00100000
00010000
000015784170
0000174840157
0000157084174
000001784157

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,157,174,157,0,0,0,0,0,84,84,0,17,0,0,0,0,17,0,84,84,0,0,0,0,0,157,174,157] >;
 

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D8E8F8G8H10A10B10C12A12B12C12D 15 20A20B20C20D24A···24H30A30B30C
order122222344444456668888888810101012121212152020202024···24303030
size111130302555566422210101010303030304441010101081212121210···10888

48 irreducible representations

dim1111111122222224444888
type++++++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6M4(2)C4×S3C4×S3C8⋊S3F5C2×F5C2×F5C4.F5S3×F5Dic3.F5C2×S3×F5
kernelC2×Dic3.F5Dic3.F5C6×C5⋊C8C2×C15⋊C8C2×D30.C2D30.C2C10×Dic3C22×D15C2×C5⋊C8C5⋊C8C2×Dic5C30Dic5C2×C10C10C2×Dic3Dic3C2×C6C6C22C2C2
# reps1411142212142281214121

In GAP, Magma, Sage, TeX 

C_2\times Dic_3.F_5
 
% in TeX
 
G:=Group("C2xDic3.F5");
 
// GroupNames label
 
G:=SmallGroup(480,1009);
 
// by ID
 
G=gap.SmallGroup(480,1009);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,120,80,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^6=d^5=1,c^2=e^4=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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