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G = C40.2D6order 480 = 25·3·5

2nd non-split extension by C40 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.2D6, D6.7D20, D20.22D6, C24.25D10, Dic6013C2, C60.99C23, Dic3.9D20, C120.24C22, Dic10.21D6, Dic30.31C22, C3⋊C8.2D10, C8.4(S3×D5), C8⋊S32D5, C40⋊C22S3, C10.9(S3×D4), C6.9(C2×D20), (C4×S3).4D10, (S3×C10).4D4, C30.24(C2×D4), C2.14(S3×D20), C51(D4.D6), C3⋊Dic2012C2, C31(C8.D10), C154(C8.C22), (C5×Dic3).4D4, (S3×Dic10)⋊10C2, D205S3.2C2, C6.D2012C2, C12.77(C22×D5), (S3×C20).27C22, C20.149(C22×S3), (C3×D20).24C22, (C3×Dic10).25C22, C4.98(C2×S3×D5), (C5×C8⋊S3)⋊2C2, (C3×C40⋊C2)⋊2C2, (C5×C3⋊C8).20C22, SmallGroup(480,350)

Series: Derived Chief Lower central Upper central

C1C60 — C40.2D6
C1C5C15C30C60C3×D20D205S3 — C40.2D6
C15C30C60 — C40.2D6
C1C2C4C8

Generators and relations for C40.2D6
 G = < a,b,c | a40=b6=1, c2=a20, bab-1=a19, cac-1=a-1, cbc-1=b-1 >

Subgroups: 684 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5, C10, C10, Dic3, Dic3 [×2], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6 [×3], C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C8.C22, C40, C40, Dic10, Dic10 [×3], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5, S3×C10, C40⋊C2, C40⋊C2, Dic20 [×2], C5×M4(2), C2×Dic10, C4○D20, D4.D6, C5×C3⋊C8, C120, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C3×Dic10, C3×D20, S3×C20, Dic30 [×2], C8.D10, C6.D20, C3⋊Dic20, C3×C40⋊C2, C5×C8⋊S3, Dic60, D205S3, S3×Dic10, C40.2D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, D4.D6, C2×S3×D5, C8.D10, S3×D20, C40.2D6

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

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B8A8B10A10B10C10D12A12B15A15B20A20B20C20D20E20F24A24B30A30B40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order1222344444556688101010101212151520202020202024243030404040404040404060606060120···120
size11620226206060222404122212124404422221212444444441212121244444···4

54 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++++-++-+-+-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D20D20C8.C22S3×D4S3×D5D4.D6C2×S3×D5C8.D10S3×D20C40.2D6
kernelC40.2D6C6.D20C3⋊Dic20C3×C40⋊C2C5×C8⋊S3Dic60D205S3S3×Dic10C40⋊C2C5×Dic3S3×C10C8⋊S3C40Dic10D20C3⋊C8C24C4×S3Dic3D6C15C10C8C5C4C3C2C1
# reps1111111111121112224411222448

Matrix representation of C40.2D6 in GL4(𝔽241) generated by

107181214121
13324025239
2712013460
21621081
,
2059636145
1483693205
20596169192
148365572
,
1515490187
3090211151
18013390187
18161211151
G:=sub<GL(4,GF(241))| [107,133,27,216,181,240,120,2,214,25,134,108,121,239,60,1],[205,148,205,148,96,36,96,36,36,93,169,55,145,205,192,72],[151,30,180,181,54,90,133,61,90,211,90,211,187,151,187,151] >;

C40.2D6 in GAP, Magma, Sage, TeX

C_{40}._2D_6
% in TeX

G:=Group("C40.2D6");
// GroupNames label

G:=SmallGroup(480,350);
// by ID

G=gap.SmallGroup(480,350);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,675,80,1356,18822]);
// Polycyclic

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

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