metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊5C4, C8⋊1Dic5, C10.4D8, C2.1D40, C20.5Q8, C10.2Q16, C2.2Dic20, C4.5Dic10, C22.9D20, (C2×C8).3D5, C5⋊3(C2.D8), (C2×C40).5C2, C20.55(C2×C4), (C2×C4).69D10, (C2×C10).14D4, C4⋊Dic5.3C2, C10.13(C4⋊C4), C4.7(C2×Dic5), C2.4(C4⋊Dic5), (C2×C20).82C22, SmallGroup(160,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊5C4
G = < a,b | a40=b4=1, bab-1=a-1 >
Smallest permutation representation of C40⋊5C4
►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)
(1 129 55 87)(2 128 56 86)(3 127 57 85)(4 126 58 84)(5 125 59 83)(6 124 60 82)(7 123 61 81)(8 122 62 120)(9 121 63 119)(10 160 64 118)(11 159 65 117)(12 158 66 116)(13 157 67 115)(14 156 68 114)(15 155 69 113)(16 154 70 112)(17 153 71 111)(18 152 72 110)(19 151 73 109)(20 150 74 108)(21 149 75 107)(22 148 76 106)(23 147 77 105)(24 146 78 104)(25 145 79 103)(26 144 80 102)(27 143 41 101)(28 142 42 100)(29 141 43 99)(30 140 44 98)(31 139 45 97)(32 138 46 96)(33 137 47 95)(34 136 48 94)(35 135 49 93)(36 134 50 92)(37 133 51 91)(38 132 52 90)(39 131 53 89)(40 130 54 88)
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), (1,129,55,87)(2,128,56,86)(3,127,57,85)(4,126,58,84)(5,125,59,83)(6,124,60,82)(7,123,61,81)(8,122,62,120)(9,121,63,119)(10,160,64,118)(11,159,65,117)(12,158,66,116)(13,157,67,115)(14,156,68,114)(15,155,69,113)(16,154,70,112)(17,153,71,111)(18,152,72,110)(19,151,73,109)(20,150,74,108)(21,149,75,107)(22,148,76,106)(23,147,77,105)(24,146,78,104)(25,145,79,103)(26,144,80,102)(27,143,41,101)(28,142,42,100)(29,141,43,99)(30,140,44,98)(31,139,45,97)(32,138,46,96)(33,137,47,95)(34,136,48,94)(35,135,49,93)(36,134,50,92)(37,133,51,91)(38,132,52,90)(39,131,53,89)(40,130,54,88)>;
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), (1,129,55,87)(2,128,56,86)(3,127,57,85)(4,126,58,84)(5,125,59,83)(6,124,60,82)(7,123,61,81)(8,122,62,120)(9,121,63,119)(10,160,64,118)(11,159,65,117)(12,158,66,116)(13,157,67,115)(14,156,68,114)(15,155,69,113)(16,154,70,112)(17,153,71,111)(18,152,72,110)(19,151,73,109)(20,150,74,108)(21,149,75,107)(22,148,76,106)(23,147,77,105)(24,146,78,104)(25,145,79,103)(26,144,80,102)(27,143,41,101)(28,142,42,100)(29,141,43,99)(30,140,44,98)(31,139,45,97)(32,138,46,96)(33,137,47,95)(34,136,48,94)(35,135,49,93)(36,134,50,92)(37,133,51,91)(38,132,52,90)(39,131,53,89)(40,130,54,88) );
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)], [(1,129,55,87),(2,128,56,86),(3,127,57,85),(4,126,58,84),(5,125,59,83),(6,124,60,82),(7,123,61,81),(8,122,62,120),(9,121,63,119),(10,160,64,118),(11,159,65,117),(12,158,66,116),(13,157,67,115),(14,156,68,114),(15,155,69,113),(16,154,70,112),(17,153,71,111),(18,152,72,110),(19,151,73,109),(20,150,74,108),(21,149,75,107),(22,148,76,106),(23,147,77,105),(24,146,78,104),(25,145,79,103),(26,144,80,102),(27,143,41,101),(28,142,42,100),(29,141,43,99),(30,140,44,98),(31,139,45,97),(32,138,46,96),(33,137,47,95),(34,136,48,94),(35,135,49,93),(36,134,50,92),(37,133,51,91),(38,132,52,90),(39,131,53,89),(40,130,54,88)]])
C40⋊5C4 is a maximal subgroup of
C40.2Q8 C10.SD32 C40.78D4 C80⋊13C4 C80⋊14C4 D40⋊7C4 C10.D16 C40.15D4 C40⋊8Q8 C40.13Q8 C4×D40 C4×Dic20 C8⋊Dic10 C42.16D10 C23.35D20 C23.10D20 C22.D40 C23.13D20 Dic5.14D8 D4.2Dic10 D10.12D8 C40⋊5C4⋊C2 Dic5.9Q16 Q8.Dic10 D10.7Q16 (C2×C8).D10 Dic10.3Q8 D20⋊4Q8 D20.3Q8 C20.7Q16 C40⋊3Q8 C8⋊(C4×D5) C40⋊2Q8 C8.6Dic10 D5×C2.D8 C8.27(C4×D5) C23.22D20 C40⋊29D4 C40.82D4 C23.47D20 C40⋊2D4 D8×Dic5 C40⋊6D4 SD16⋊Dic5 C40⋊8D4 Q16×Dic5 D10⋊3Q16 C60.5Q8 C120⋊9C4
C40⋊5C4 is a maximal quotient of
C40⋊5C8 C80⋊13C4 C80⋊14C4 C80.6C4 C20.39C42 C60.5Q8 C120⋊9C4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | + | - | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D5 | D8 | Q16 | Dic5 | D10 | Dic10 | D20 | D40 | Dic20 |
| kernel | C40⋊5C4 | C4⋊Dic5 | C2×C40 | C40 | C20 | C2×C10 | C2×C8 | C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C40⋊5C4 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 39 | 38 |
| 0 | 24 | 15 |
| 32 | 0 | 0 |
| 0 | 40 | 1 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(41))| [1,0,0,0,39,24,0,38,15],[32,0,0,0,40,0,0,1,1] >;
C40⋊5C4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_5C_4 % in TeX
G:=Group("C40:5C4"); // GroupNames label
G:=SmallGroup(160,25);
// by ID
G=gap.SmallGroup(160,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,4613]);
// Polycyclic
G:=Group<a,b|a^40=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export