Copied to
clipboard

G = C40.55D6order 480 = 25·3·5

16th non-split extension by C40 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.55D6, C24.37D10, C60.172C23, C120.59C22, (S3×C8)⋊8D5, (S3×C40)⋊9C2, C15⋊Q8.4C4, C8⋊D57S3, C159(C8○D4), C3⋊C8.36D10, D6.4(C4×D5), C8.15(S3×D5), C54(D12.C4), (C4×D5).56D6, D10.5(C4×S3), C52C8.23D6, C5⋊D12.4C4, C15⋊D4.4C4, C3⋊D20.4C4, C40⋊S312C2, D30.19(C2×C4), (C4×S3).44D10, Dic3.6(C4×D5), Dic5.5(C4×S3), D152C810C2, C32(D20.3C4), D6.Dic511C2, C30.38(C22×C4), (S3×C20).52C22, C20.169(C22×S3), C153C8.28C22, Dic15.20(C2×C4), D6.D10.2C2, (D5×C12).56C22, (C4×D15).39C22, C12.169(C22×D5), (D5×C3⋊C8)⋊9C2, C6.7(C2×C4×D5), C2.10(C4×S3×D5), C10.38(S3×C2×C4), C4.142(C2×S3×D5), (C3×C8⋊D5)⋊9C2, (C6×D5).3(C2×C4), (C5×C3⋊C8).38C22, (S3×C10).27(C2×C4), (C3×Dic5).3(C2×C4), (C3×C52C8).23C22, (C5×Dic3).32(C2×C4), SmallGroup(480,343)

Series: Derived Chief Lower central Upper central

C1C30 — C40.55D6
C1C5C15C30C60D5×C12D6.D10 — C40.55D6
C15C30 — C40.55D6
C1C4C8

Generators and relations for C40.55D6
 G = < a,b,c | a40=b6=1, c2=a20, bab-1=cac-1=a29, cbc-1=a20b-1 >

Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4 [×2], C2×C20, S3×C8, S3×C8, C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5 [×2], C8⋊D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D12.C4, C5×C3⋊C8, C3×C52C8, C153C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.3C4, D5×C3⋊C8, D152C8, D6.Dic5, C3×C8⋊D5, S3×C40, C40⋊S3, D6.D10, C40.55D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D12.C4, C2×S3×D5, D20.3C4, C4×S3×D5, C40.55D6

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B8A8B8C8D8E8F8G8H8I8J10A10B10C10D10E10F12A12B12C15A15B20A20B20C20D20E20F20G20H24A24B24C24D30A30B40A···40H40I···40P60A60B60C60D120A···120H
order12222344444556688888888881010101010101212121515202020202020202024242424303040···4040···4060606060120···120
size116103021161030222202233331010303022666622204422226666442020442···26···644444···4

78 irreducible representations

dim1111111111112222222222222244444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D5D6D6D6D10D10D10C4×S3C4×S3C8○D4C4×D5C4×D5D20.3C4S3×D5D12.C4C2×S3×D5C4×S3×D5C40.55D6
kernelC40.55D6D5×C3⋊C8D152C8D6.Dic5C3×C8⋊D5S3×C40C40⋊S3D6.D10C15⋊D4C3⋊D20C5⋊D12C15⋊Q8C8⋊D5S3×C8C52C8C40C4×D5C3⋊C8C24C4×S3Dic5D10C15Dic3D6C3C8C5C4C2C1
# reps11111111222212111222224441622248

Matrix representation of C40.55D6 in GL4(𝔽241) generated by

707000
17111500
001770
000177
,
15100
024000
0023949
001771
,
766900
4916500
0035115
00113206
G:=sub<GL(4,GF(241))| [70,171,0,0,70,115,0,0,0,0,177,0,0,0,0,177],[1,0,0,0,51,240,0,0,0,0,239,177,0,0,49,1],[76,49,0,0,69,165,0,0,0,0,35,113,0,0,115,206] >;

C40.55D6 in GAP, Magma, Sage, TeX

C_{40}._{55}D_6
% in TeX

G:=Group("C40.55D6");
// GroupNames label

G:=SmallGroup(480,343);
// by ID

G=gap.SmallGroup(480,343);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^29,c*b*c^-1=a^20*b^-1>;
// generators/relations

׿
×
𝔽