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G = C40.52D6order 480 = 25·3·5

13rd non-split extension by C40 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.52D6, C156M5(2), C24.59D10, C120.55C22, C52C164S3, C55(D6.C8), D6.(C52C8), (S3×C8).2D5, C8.38(S3×D5), C3⋊C8.2Dic5, (S3×C40).3C2, (S3×C10).4C8, (S3×C20).9C4, C10.21(S3×C8), C30.23(C2×C8), C153C1612C2, C31(C20.4C8), C20.106(C4×S3), C60.139(C2×C4), Dic3.(C52C8), (C4×S3).2Dic5, (C5×Dic3).4C8, C4.17(S3×Dic5), C12.22(C2×Dic5), (C5×C3⋊C8).7C4, C6.2(C2×C52C8), C2.3(S3×C52C8), (C3×C52C16)⋊9C2, SmallGroup(480,11)

Series: Derived Chief Lower central Upper central

C1C30 — C40.52D6
C1C5C15C30C60C120C3×C52C16 — C40.52D6
C15C30 — C40.52D6
C1C8

Generators and relations for C40.52D6
 G = < a,b,c | a40=1, b6=a35, c2=a5, bab-1=cac-1=a9, cbc-1=a10b5 >

6C2
3C22
3C4
2S3
6C10
3C8
3C2×C4
3C20
3C2×C10
2C5×S3
3C2×C8
5C16
15C16
3C40
3C2×C20
15M5(2)
5C48
5C3⋊C16
3C2×C40
3C52C16
5D6.C8
3C20.4C8

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

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

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

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

84 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B 6 8A8B8C8D8E8F10A10B10C10D10E10F12A12B15A15B16A16B16C16D16E16F16G16H20A20B20C20D20E20F20G20H24A24B24C24D30A30B40A···40H40I···40P48A···48H60A60B60C60D120A···120H
order1223444556888888101010101010121215151616161616161616202020202020202024242424303040···4040···4048···4860606060120···120
size116211622211116622666622441010101030303030222266662222442···26···610···1044444···4

84 irreducible representations

dim1111111122222222222224444
type+++++++-+-+-
imageC1C2C2C2C4C4C8C8S3D5D6Dic5D10Dic5C4×S3M5(2)C52C8C52C8S3×C8D6.C8C20.4C8S3×D5S3×Dic5S3×C52C8C40.52D6
kernelC40.52D6C3×C52C16C153C16S3×C40C5×C3⋊C8S3×C20C5×Dic3S3×C10C52C16S3×C8C40C3⋊C8C24C4×S3C20C15Dic3D6C10C5C3C8C4C2C1
# reps11112244121222244448162248

Matrix representation of C40.52D6 in GL4(𝔽241) generated by

8000
0800
001800
0013236
,
03300
1398200
00240144
00151
,
020800
139000
00197
0092240
G:=sub<GL(4,GF(241))| [8,0,0,0,0,8,0,0,0,0,180,13,0,0,0,236],[0,139,0,0,33,82,0,0,0,0,240,15,0,0,144,1],[0,139,0,0,208,0,0,0,0,0,1,92,0,0,97,240] >;

C40.52D6 in GAP, Magma, Sage, TeX

C_{40}._{52}D_6
% in TeX

G:=Group("C40.52D6");
// GroupNames label

G:=SmallGroup(480,11);
// by ID

G=gap.SmallGroup(480,11);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^6=a^35,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^5>;
// generators/relations

Export

Subgroup lattice of C40.52D6 in TeX

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