metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8C4, C8⋊3Dic5, C10.9C42, C10.7M4(2), C5⋊2C8⋊4C4, (C2×C8).8D5, C5⋊4(C8⋊C4), C4.21(C4×D5), C20.62(C2×C4), (C2×C40).12C2, (C2×C4).92D10, C2.4(C4×Dic5), C2.2(C8⋊D5), (C4×Dic5).6C2, (C2×Dic5).4C4, C4.13(C2×Dic5), C22.10(C4×D5), (C2×C20).106C22, (C2×C5⋊2C8).10C2, (C2×C10).31(C2×C4), SmallGroup(160,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8C4
G = < a,b | a40=b4=1, bab-1=a29 >
Smallest permutation representation of C40⋊8C4
►Regular action on 160 points
Generators in S160
(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)
(1 74 123 113)(2 63 124 102)(3 52 125 91)(4 41 126 120)(5 70 127 109)(6 59 128 98)(7 48 129 87)(8 77 130 116)(9 66 131 105)(10 55 132 94)(11 44 133 83)(12 73 134 112)(13 62 135 101)(14 51 136 90)(15 80 137 119)(16 69 138 108)(17 58 139 97)(18 47 140 86)(19 76 141 115)(20 65 142 104)(21 54 143 93)(22 43 144 82)(23 72 145 111)(24 61 146 100)(25 50 147 89)(26 79 148 118)(27 68 149 107)(28 57 150 96)(29 46 151 85)(30 75 152 114)(31 64 153 103)(32 53 154 92)(33 42 155 81)(34 71 156 110)(35 60 157 99)(36 49 158 88)(37 78 159 117)(38 67 160 106)(39 56 121 95)(40 45 122 84)
G:=sub<Sym(160)| (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), (1,74,123,113)(2,63,124,102)(3,52,125,91)(4,41,126,120)(5,70,127,109)(6,59,128,98)(7,48,129,87)(8,77,130,116)(9,66,131,105)(10,55,132,94)(11,44,133,83)(12,73,134,112)(13,62,135,101)(14,51,136,90)(15,80,137,119)(16,69,138,108)(17,58,139,97)(18,47,140,86)(19,76,141,115)(20,65,142,104)(21,54,143,93)(22,43,144,82)(23,72,145,111)(24,61,146,100)(25,50,147,89)(26,79,148,118)(27,68,149,107)(28,57,150,96)(29,46,151,85)(30,75,152,114)(31,64,153,103)(32,53,154,92)(33,42,155,81)(34,71,156,110)(35,60,157,99)(36,49,158,88)(37,78,159,117)(38,67,160,106)(39,56,121,95)(40,45,122,84)>;
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), (1,74,123,113)(2,63,124,102)(3,52,125,91)(4,41,126,120)(5,70,127,109)(6,59,128,98)(7,48,129,87)(8,77,130,116)(9,66,131,105)(10,55,132,94)(11,44,133,83)(12,73,134,112)(13,62,135,101)(14,51,136,90)(15,80,137,119)(16,69,138,108)(17,58,139,97)(18,47,140,86)(19,76,141,115)(20,65,142,104)(21,54,143,93)(22,43,144,82)(23,72,145,111)(24,61,146,100)(25,50,147,89)(26,79,148,118)(27,68,149,107)(28,57,150,96)(29,46,151,85)(30,75,152,114)(31,64,153,103)(32,53,154,92)(33,42,155,81)(34,71,156,110)(35,60,157,99)(36,49,158,88)(37,78,159,117)(38,67,160,106)(39,56,121,95)(40,45,122,84) );
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)], [(1,74,123,113),(2,63,124,102),(3,52,125,91),(4,41,126,120),(5,70,127,109),(6,59,128,98),(7,48,129,87),(8,77,130,116),(9,66,131,105),(10,55,132,94),(11,44,133,83),(12,73,134,112),(13,62,135,101),(14,51,136,90),(15,80,137,119),(16,69,138,108),(17,58,139,97),(18,47,140,86),(19,76,141,115),(20,65,142,104),(21,54,143,93),(22,43,144,82),(23,72,145,111),(24,61,146,100),(25,50,147,89),(26,79,148,118),(27,68,149,107),(28,57,150,96),(29,46,151,85),(30,75,152,114),(31,64,153,103),(32,53,154,92),(33,42,155,81),(34,71,156,110),(35,60,157,99),(36,49,158,88),(37,78,159,117),(38,67,160,106),(39,56,121,95),(40,45,122,84)]])
C40⋊8C4 is a maximal subgroup of
C80⋊C4 C20.23C42 C40⋊11Q8 C4×C8⋊D5 D10.5C42 C40⋊Q8 D5×C8⋊C4 D10.7C42 Dic5.9M4(2) C40⋊8C4⋊C2 D10⋊4M4(2) C5⋊2C8⋊26D4 D4.D5⋊5C4 C4⋊C4.D10 C20⋊Q8⋊C2 D4⋊D5⋊6C4 C5⋊Q16⋊5C4 Q8⋊C4⋊D5 C40⋊8C4.C2 Q8⋊D5⋊6C4 Dic5.5M4(2) C42.198D10 C42.202D10 C42.31D10 Dic20⋊15C4 C40⋊3Q8 D40⋊15C4 C40⋊4Q8 C40⋊21(C2×C4) D40⋊16C4 C20.42C42 C40⋊32D4 M4(2)×Dic5 C20.37C42 C40⋊D4 C40.50D4 D8⋊Dic5 C40⋊11D4 SD16⋊Dic5 C40.31D4 C40⋊9D4 Q16⋊Dic5 C40.37D4 D8⋊4Dic5 C30.21C42 C30.23C42 C120⋊13C4
C40⋊8C4 is a maximal quotient of
C42.279D10 C40⋊8C8 C80⋊C4 (C2×C40)⋊15C4 C30.21C42 C30.23C42 C120⋊13C4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | Dic5 | D10 | C4×D5 | C4×D5 | C8⋊D5 |
| kernel | C40⋊8C4 | C2×C5⋊2C8 | C4×Dic5 | C2×C40 | C5⋊2C8 | C40 | C2×Dic5 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C40⋊8C4 ►in GL4(𝔽41) generated by
| 6 | 35 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 13 | 22 |
| 0 | 0 | 19 | 22 |
| 6 | 18 | 0 | 0 |
| 23 | 35 | 0 | 0 |
| 0 | 0 | 2 | 15 |
| 0 | 0 | 27 | 39 |
G:=sub<GL(4,GF(41))| [6,6,0,0,35,1,0,0,0,0,13,19,0,0,22,22],[6,23,0,0,18,35,0,0,0,0,2,27,0,0,15,39] >;
C40⋊8C4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8C_4 % in TeX
G:=Group("C40:8C4"); // GroupNames label
G:=SmallGroup(160,22);
// by ID
G=gap.SmallGroup(160,22);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,69,4613]);
// Polycyclic
G:=Group<a,b|a^40=b^4=1,b*a*b^-1=a^29>;
// generators/relations
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