Copied to
clipboard

## G = C5×Q8.13D6order 480 = 25·3·5

### Direct product of C5 and Q8.13D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8.13D6
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×C4○D12 — C5×Q8.13D6
 Lower central C3 — C6 — C12 — C5×Q8.13D6
 Upper central C1 — C20 — C2×C20 — C5×C4○D4

Generators and relations for C5×Q8.13D6
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, cd=dc, ece-1=b-1c, ede-1=d5 >

Subgroups: 276 in 124 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C10, C10 [×3], Dic3, C12 [×2], C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C5×S3, C30, C30 [×2], C4○D8, C40 [×2], C2×C20, C2×C20 [×2], C5×D4, C5×D4 [×3], C5×Q8, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C60 [×2], C60, S3×C10, C2×C30, C2×C30, C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4, C5×C4○D4, Q8.13D6, C5×C3⋊C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4○D8, C10×C3⋊C8, C5×D4⋊S3, C5×D4.S3, C5×Q82S3, C5×C3⋊Q16, C5×C4○D12, C15×C4○D4, C5×Q8.13D6
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, C4○D8, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8.13D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C4○D8, C10×C3⋊D4, C5×Q8.13D6

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 10O 10P 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 20Q 20R 20S 20T 30A 30B 30C 30D 30E ··· 30P 40A ··· 40P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 6 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 size 1 1 2 4 12 2 1 1 2 4 12 1 1 1 1 2 4 4 4 6 6 6 6 1 1 1 1 2 2 2 2 4 4 4 4 12 12 12 12 2 2 4 4 4 2 2 2 2 1 ··· 1 2 2 2 2 4 4 4 4 12 12 12 12 2 2 2 2 4 ··· 4 6 ··· 6 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C5×S3 C4○D8 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C5×C3⋊D4 C5×C3⋊D4 C5×C4○D8 Q8.13D6 C5×Q8.13D6 kernel C5×Q8.13D6 C10×C3⋊C8 C5×D4⋊S3 C5×D4.S3 C5×Q8⋊2S3 C5×C3⋊Q16 C5×C4○D12 C15×C4○D4 Q8.13D6 C2×C3⋊C8 D4⋊S3 D4.S3 Q8⋊2S3 C3⋊Q16 C4○D12 C3×C4○D4 C5×C4○D4 C60 C2×C30 C2×C20 C5×D4 C5×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C3 C5 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 8 8 16 2 8

Matrix representation of C5×Q8.13D6 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 98 0 0 0 0 98
,
 240 0 0 0 0 240 0 0 0 0 64 0 0 0 64 177
,
 171 101 0 0 140 70 0 0 0 0 177 128 0 0 0 64
,
 0 240 0 0 1 1 0 0 0 0 177 0 0 0 0 177
,
 239 202 0 0 204 2 0 0 0 0 211 60 0 0 230 30
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[240,0,0,0,0,240,0,0,0,0,64,64,0,0,0,177],[171,140,0,0,101,70,0,0,0,0,177,0,0,0,128,64],[0,1,0,0,240,1,0,0,0,0,177,0,0,0,0,177],[239,204,0,0,202,2,0,0,0,0,211,230,0,0,60,30] >;

C5×Q8.13D6 in GAP, Magma, Sage, TeX

C_5\times Q_8._{13}D_6
% in TeX

G:=Group("C5xQ8.13D6");
// GroupNames label

G:=SmallGroup(480,829);
// by ID

G=gap.SmallGroup(480,829);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,4204,1068,102,15686]);
// 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,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations

׿
×
𝔽