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G = C40.D6order 480 = 25·3·5

6th non-split extension by C40 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.6D6, C30.4D8, C151SD32, D40.2S3, C60.73D4, C24.6D10, Dic123D5, C120.33C22, C52(D8.S3), C8.26(S3×D5), C153C165C2, C6.9(D4⋊D5), (C3×D40).3C2, C32(C5⋊SD32), C10.9(D4⋊S3), (C5×Dic12)⋊5C2, C12.2(C5⋊D4), C20.2(C3⋊D4), C2.5(C15⋊D8), C4.2(C15⋊D4), SmallGroup(480,16)

Series: Derived Chief Lower central Upper central

C1C120 — C40.D6
C1C5C15C30C60C120C3×D40 — C40.D6
C15C30C60C120 — C40.D6
C1C2C4C8

Generators and relations for C40.D6
 G = < a,b,c | a40=b6=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a25b-1 >

40C2
12C4
20C22
40C6
8D5
6Q8
10D4
4Dic3
20C2×C6
4D10
12C20
8C3×D5
3Q16
5D8
15C16
2Dic6
10C3×D4
2D20
6C5×Q8
4C6×D5
4C5×Dic3
15SD32
5C3×D8
5C3⋊C16
3C5×Q16
3C52C16
2C5×Dic6
2C3×D20
5D8.S3
3C5⋊SD32

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

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

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

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

48 conjugacy classes

class 1 2A2B 3 4A4B5A5B6A6B6C8A8B10A10B 12 15A15B16A16B16C16D20A20B20C20D20E20F24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order1223445566688101012151516161616202020202020242430304040404060606060120···120
size11402224222404022224443030303044242424244444444444444···4

48 irreducible representations

dim111122222222244444444
type+++++++++++++--+
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4SD32C5⋊D4D4⋊S3S3×D5D4⋊D5D8.S3C15⋊D4C5⋊SD32C15⋊D8C40.D6
kernelC40.D6C153C16C3×D40C5×Dic12D40C60Dic12C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps111111212224412222448

Matrix representation of C40.D6 in GL6(𝔽241)

791290000
231840000
005118900
0051000
000010
000001
,
100000
2252400000
0051100
005119000
00002250
000011615
,
482220000
621110000
0051100
005119000
000012483
0000169117

G:=sub<GL(6,GF(241))| [79,23,0,0,0,0,129,184,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,225,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,225,116,0,0,0,0,0,15],[48,62,0,0,0,0,222,111,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,124,169,0,0,0,0,83,117] >;

C40.D6 in GAP, Magma, Sage, TeX

C_{40}.D_6
% in TeX

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

G:=SmallGroup(480,16);
// by ID

G=gap.SmallGroup(480,16);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,120,254,135,142,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^25*b^-1>;
// generators/relations

Export

Subgroup lattice of C40.D6 in TeX

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