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G = C5×Q8.11D6order 480 = 25·3·5

Direct product of C5 and Q8.11D6

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

Aliases: C5×Q8.11D6, C60.152D4, C60.231C23, (C6×Q8)⋊2C10, (Q8×C10)⋊13S3, (Q8×C30)⋊16C2, C3⋊Q165C10, C6.54(D4×C10), C12.19(C5×D4), (C5×Q8).55D6, Q82S35C10, C4○D12.5C10, C30.437(C2×D4), (C2×C20).245D6, (C2×C30).185D4, C4.Dic37C10, Q8.11(S3×C10), D12.10(C2×C10), C20.96(C3⋊D4), C1536(C8.C22), Dic6.9(C2×C10), C20.204(C22×S3), (C2×C60).365C22, C12.15(C22×C10), (C5×D12).49C22, (Q8×C15).49C22, (C5×Dic6).51C22, C3⋊C8.3(C2×C10), C4.15(S3×C2×C10), (C2×Q8)⋊4(C5×S3), C34(C5×C8.C22), (C2×C6).42(C5×D4), C4.17(C5×C3⋊D4), (C2×C4).18(S3×C10), (C5×C3⋊Q16)⋊13C2, C2.18(C10×C3⋊D4), (C5×C3⋊C8).29C22, (C3×Q8).6(C2×C10), (C2×C12).38(C2×C10), (C5×Q82S3)⋊13C2, (C5×C4○D12).11C2, C10.139(C2×C3⋊D4), (C5×C4.Dic3)⋊19C2, C22.11(C5×C3⋊D4), (C2×C10).64(C3⋊D4), SmallGroup(480,821)

Series: Derived Chief Lower central Upper central

C1C12 — C5×Q8.11D6
C1C3C6C12C60C5×D12C5×C4○D12 — C5×Q8.11D6
C3C6C12 — C5×Q8.11D6
C1C10C2×C20Q8×C10

Generators and relations for C5×Q8.11D6
 G = < a,b,c,d,e | a5=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >

Subgroups: 260 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], C10, C10 [×2], Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C5×S3, C30, C30, C8.C22, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C5×Q8 [×2], C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, C5×Dic3, C60 [×2], C60 [×2], S3×C10, C2×C30, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], Q8×C10, C5×C4○D4, Q8.11D6, C5×C3⋊C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C5×C8.C22, C5×C4.Dic3, C5×Q82S3 [×2], C5×C3⋊Q16 [×2], C5×C4○D12, Q8×C30, C5×Q8.11D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C8.C22, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8.11D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C8.C22, C10×C3⋊D4, C5×Q8.11D6

Smallest permutation representation of C5×Q8.11D6
On 240 points
Generators in S240
(1 56 44 35 15)(2 57 45 36 16)(3 58 46 25 17)(4 59 47 26 18)(5 60 48 27 19)(6 49 37 28 20)(7 50 38 29 21)(8 51 39 30 22)(9 52 40 31 23)(10 53 41 32 24)(11 54 42 33 13)(12 55 43 34 14)(61 119 97 95 73)(62 120 98 96 74)(63 109 99 85 75)(64 110 100 86 76)(65 111 101 87 77)(66 112 102 88 78)(67 113 103 89 79)(68 114 104 90 80)(69 115 105 91 81)(70 116 106 92 82)(71 117 107 93 83)(72 118 108 94 84)(121 178 160 145 133)(122 179 161 146 134)(123 180 162 147 135)(124 169 163 148 136)(125 170 164 149 137)(126 171 165 150 138)(127 172 166 151 139)(128 173 167 152 140)(129 174 168 153 141)(130 175 157 154 142)(131 176 158 155 143)(132 177 159 156 144)(181 238 226 208 196)(182 239 227 209 197)(183 240 228 210 198)(184 229 217 211 199)(185 230 218 212 200)(186 231 219 213 201)(187 232 220 214 202)(188 233 221 215 203)(189 234 222 216 204)(190 235 223 205 193)(191 236 224 206 194)(192 237 225 207 195)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 130 127 124)(122 131 128 125)(123 132 129 126)(133 142 139 136)(134 143 140 137)(135 144 141 138)(145 154 151 148)(146 155 152 149)(147 156 153 150)(157 166 163 160)(158 167 164 161)(159 168 165 162)(169 178 175 172)(170 179 176 173)(171 180 177 174)(181 190 187 184)(182 191 188 185)(183 192 189 186)(193 202 199 196)(194 203 200 197)(195 204 201 198)(205 214 211 208)(206 215 212 209)(207 216 213 210)(217 226 223 220)(218 227 224 221)(219 228 225 222)(229 238 235 232)(230 239 236 233)(231 240 237 234)
(1 65 7 71)(2 72 8 66)(3 67 9 61)(4 62 10 68)(5 69 11 63)(6 64 12 70)(13 75 19 81)(14 82 20 76)(15 77 21 83)(16 84 22 78)(17 79 23 73)(18 74 24 80)(25 89 31 95)(26 96 32 90)(27 91 33 85)(28 86 34 92)(29 93 35 87)(30 88 36 94)(37 100 43 106)(38 107 44 101)(39 102 45 108)(40 97 46 103)(41 104 47 98)(42 99 48 105)(49 110 55 116)(50 117 56 111)(51 112 57 118)(52 119 58 113)(53 114 59 120)(54 109 60 115)(121 181 127 187)(122 188 128 182)(123 183 129 189)(124 190 130 184)(125 185 131 191)(126 192 132 186)(133 196 139 202)(134 203 140 197)(135 198 141 204)(136 193 142 199)(137 200 143 194)(138 195 144 201)(145 208 151 214)(146 215 152 209)(147 210 153 216)(148 205 154 211)(149 212 155 206)(150 207 156 213)(157 217 163 223)(158 224 164 218)(159 219 165 225)(160 226 166 220)(161 221 167 227)(162 228 168 222)(169 235 175 229)(170 230 176 236)(171 237 177 231)(172 232 178 238)(173 239 179 233)(174 234 180 240)
(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)
(1 132 7 126)(2 125 8 131)(3 130 9 124)(4 123 10 129)(5 128 11 122)(6 121 12 127)(13 134 19 140)(14 139 20 133)(15 144 21 138)(16 137 22 143)(17 142 23 136)(18 135 24 141)(25 154 31 148)(26 147 32 153)(27 152 33 146)(28 145 34 151)(29 150 35 156)(30 155 36 149)(37 160 43 166)(38 165 44 159)(39 158 45 164)(40 163 46 157)(41 168 47 162)(42 161 48 167)(49 178 55 172)(50 171 56 177)(51 176 57 170)(52 169 58 175)(53 174 59 180)(54 179 60 173)(61 181 67 187)(62 186 68 192)(63 191 69 185)(64 184 70 190)(65 189 71 183)(66 182 72 188)(73 196 79 202)(74 201 80 195)(75 194 81 200)(76 199 82 193)(77 204 83 198)(78 197 84 203)(85 206 91 212)(86 211 92 205)(87 216 93 210)(88 209 94 215)(89 214 95 208)(90 207 96 213)(97 226 103 220)(98 219 104 225)(99 224 105 218)(100 217 106 223)(101 222 107 228)(102 227 108 221)(109 236 115 230)(110 229 116 235)(111 234 117 240)(112 239 118 233)(113 232 119 238)(114 237 120 231)

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

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

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

105 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B5C5D6A6B6C8A8B10A10B10C10D10E10F10G10H10I10J10K10L12A···12F15A15B15C15D20A···20H20I···20P20Q20R20S20T30A···30L40A···40H60A···60X
order122234444455556668810101010101010101010101012···121515151520···2020···202020202030···3040···4060···60
size1121222244121111222121211112222121212124···422222···24···4121212122···212···124···4

105 irreducible representations

dim111111111111222222222222224444
type+++++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6C3⋊D4C3⋊D4C5×S3C5×D4C5×D4S3×C10S3×C10C5×C3⋊D4C5×C3⋊D4C8.C22Q8.11D6C5×C8.C22C5×Q8.11D6
kernelC5×Q8.11D6C5×C4.Dic3C5×Q82S3C5×C3⋊Q16C5×C4○D12Q8×C30Q8.11D6C4.Dic3Q82S3C3⋊Q16C4○D12C6×Q8Q8×C10C60C2×C30C2×C20C5×Q8C20C2×C10C2×Q8C12C2×C6C2×C4Q8C4C22C15C5C3C1
# reps112211448844111122244448881248

Matrix representation of C5×Q8.11D6 in GL6(𝔽241)

100000
010000
0091000
0009100
0000910
0000091
,
24000000
02400000
000100
00240000
001231832402
00153302401
,
1711010000
140700000
002221900
00191900
007313838203
0022612319203
,
010000
2402400000
00024000
001000
001231832402
002111532401
,
1512300000
79900000
001231832402
000010
00024000
0014772211118

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,91,0,0,0,0,0,0,91,0,0,0,0,0,0,91,0,0,0,0,0,0,91],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,123,153,0,0,1,0,183,30,0,0,0,0,240,240,0,0,0,0,2,1],[171,140,0,0,0,0,101,70,0,0,0,0,0,0,222,19,73,226,0,0,19,19,138,123,0,0,0,0,38,19,0,0,0,0,203,203],[0,240,0,0,0,0,1,240,0,0,0,0,0,0,0,1,123,211,0,0,240,0,183,153,0,0,0,0,240,240,0,0,0,0,2,1],[151,79,0,0,0,0,230,90,0,0,0,0,0,0,123,0,0,147,0,0,183,0,240,72,0,0,240,1,0,211,0,0,2,0,0,118] >;

C5×Q8.11D6 in GAP, Magma, Sage, TeX

C_5\times Q_8._{11}D_6
% in TeX

G:=Group("C5xQ8.11D6");
// GroupNames label

G:=SmallGroup(480,821);
// by ID

G=gap.SmallGroup(480,821);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,436,4204,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations

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