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G = C40.31D6order 480 = 25·3·5

31st non-split extension by C40 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.31D6, D10.5D12, C24.47D10, D12.21D10, C60.123C23, C120.31C22, Dic5.24D12, Dic6.21D10, D60.33C22, Dic30.35C22, (C8×D5)⋊5S3, C6.8(D4×D5), (D5×C24)⋊5C2, C24⋊C27D5, C51(C4○D24), C151(C4○D8), C8.23(S3×D5), C5⋊D2412C2, (C6×D5).43D4, (C4×D5).79D6, C10.8(C2×D12), C30.19(C2×D4), C2.13(D5×D12), C52C8.32D6, C24⋊D514C2, D125D59C2, C12.28D109C2, C5⋊Dic1212C2, C31(SD163D5), C20.76(C22×S3), (C3×Dic5).47D4, (D5×C12).93C22, (C5×D12).23C22, C12.146(C22×D5), (C5×Dic6).25C22, C4.71(C2×S3×D5), (C5×C24⋊C2)⋊4C2, (C3×C52C8).36C22, SmallGroup(480,345)

Series: Derived Chief Lower central Upper central

C1C60 — C40.31D6
C1C5C15C30C60D5×C12D125D5 — C40.31D6
C15C30C60 — C40.31D6
C1C2C4C8

Generators and relations for C40.31D6
 G = < a,b,c | a40=b6=1, c2=a20, bab-1=a9, cac-1=a19, cbc-1=a20b-1 >

Subgroups: 764 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, Dic6, Dic6, C4×S3 [×2], D12, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, C4○D8, C52C8, C40, Dic10, C4×D5, C4×D5, D20 [×2], C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C24⋊C2, C24⋊C2, D24, Dic12, C2×C24, C4○D12 [×2], C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, C4○D24, C3×C52C8, C120, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, D5×C12, C5×Dic6, C5×D12, Dic30, D60, SD163D5, C5⋊D24, C5⋊Dic12, D5×C24, C5×C24⋊C2, C24⋊D5, D125D5, C12.28D10, C40.31D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C4○D8, C22×D5, C2×D12, S3×D5, D4×D5, C4○D24, C2×S3×D5, SD163D5, D5×D12, C40.31D6

Smallest permutation representation of C40.31D6
On 240 points
Generators in S240
(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)
(1 148 224)(2 157 225 10 149 233)(3 126 226 19 150 202)(4 135 227 28 151 211)(5 144 228 37 152 220)(6 153 229)(7 122 230 15 154 238)(8 131 231 24 155 207)(9 140 232 33 156 216)(11 158 234)(12 127 235 20 159 203)(13 136 236 29 160 212)(14 145 237 38 121 221)(16 123 239)(17 132 240 25 124 208)(18 141 201 34 125 217)(21 128 204)(22 137 205 30 129 213)(23 146 206 39 130 222)(26 133 209)(27 142 210 35 134 218)(31 138 214)(32 147 215 40 139 223)(36 143 219)(41 161 119 61 181 99)(42 170 120 70 182 108)(43 179 81 79 183 117)(44 188 82 48 184 86)(45 197 83 57 185 95)(46 166 84 66 186 104)(47 175 85 75 187 113)(49 193 87 53 189 91)(50 162 88 62 190 100)(51 171 89 71 191 109)(52 180 90 80 192 118)(54 198 92 58 194 96)(55 167 93 67 195 105)(56 176 94 76 196 114)(59 163 97 63 199 101)(60 172 98 72 200 110)(64 168 102 68 164 106)(65 177 103 77 165 115)(69 173 107 73 169 111)(74 178 112 78 174 116)
(1 114 21 94)(2 93 22 113)(3 112 23 92)(4 91 24 111)(5 110 25 90)(6 89 26 109)(7 108 27 88)(8 87 28 107)(9 106 29 86)(10 85 30 105)(11 104 31 84)(12 83 32 103)(13 102 33 82)(14 81 34 101)(15 100 35 120)(16 119 36 99)(17 98 37 118)(18 117 38 97)(19 96 39 116)(20 115 40 95)(41 219 61 239)(42 238 62 218)(43 217 63 237)(44 236 64 216)(45 215 65 235)(46 234 66 214)(47 213 67 233)(48 232 68 212)(49 211 69 231)(50 230 70 210)(51 209 71 229)(52 228 72 208)(53 207 73 227)(54 226 74 206)(55 205 75 225)(56 224 76 204)(57 203 77 223)(58 222 78 202)(59 201 79 221)(60 220 80 240)(121 183 141 163)(122 162 142 182)(123 181 143 161)(124 200 144 180)(125 179 145 199)(126 198 146 178)(127 177 147 197)(128 196 148 176)(129 175 149 195)(130 194 150 174)(131 173 151 193)(132 192 152 172)(133 171 153 191)(134 190 154 170)(135 169 155 189)(136 188 156 168)(137 167 157 187)(138 186 158 166)(139 165 159 185)(140 184 160 164)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C8A8B8C8D10A10B10C10D12A12B12C12D15A15B20A20B20C20D24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order122223444445566688881010101012121212151520202020242424242424242430304040404060606060120···120
size111012602255126022210102210102224242210104444242422221010101044444444444···4

60 irreducible representations

dim1111111122222222222222444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D12D12C4○D8C4○D24S3×D5D4×D5C2×S3×D5SD163D5D5×D12C40.31D6
kernelC40.31D6C5⋊D24C5⋊Dic12D5×C24C5×C24⋊C2C24⋊D5D125D5C12.28D10C8×D5C3×Dic5C6×D5C24⋊C2C52C8C40C4×D5C24Dic6D12Dic5D10C15C5C8C6C4C3C2C1
# reps1111111111121112222248222448

Matrix representation of C40.31D6 in GL6(𝔽241)

56620000
1011850000
00233000
006121100
000019052
00001900
,
1490000
1772390000
001000
0013724000
0000511
000051190
,
621350000
841790000
0013723900
0022710400
0000190240
000019051

G:=sub<GL(6,GF(241))| [56,101,0,0,0,0,62,185,0,0,0,0,0,0,233,61,0,0,0,0,0,211,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[1,177,0,0,0,0,49,239,0,0,0,0,0,0,1,137,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,1,190],[62,84,0,0,0,0,135,179,0,0,0,0,0,0,137,227,0,0,0,0,239,104,0,0,0,0,0,0,190,190,0,0,0,0,240,51] >;

C40.31D6 in GAP, Magma, Sage, TeX

C_{40}._{31}D_6
% in TeX

G:=Group("C40.31D6");
// GroupNames label

G:=SmallGroup(480,345);
// by ID

G=gap.SmallGroup(480,345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,58,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^19,c*b*c^-1=a^20*b^-1>;
// generators/relations

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