metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6C4, C8⋊2Dic5, C20.4Q8, C4.4Dic10, C10.2SD16, C22.8D20, (C2×C8).6D5, C5⋊3(C4.Q8), (C2×C40).8C2, C20.54(C2×C4), (C2×C10).13D4, (C2×C4).68D10, C4⋊Dic5.2C2, C10.12(C4⋊C4), C4.6(C2×Dic5), C2.2(C40⋊C2), C2.3(C4⋊Dic5), (C2×C20).81C22, SmallGroup(160,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊6C4
G = < a,b | a40=b4=1, bab-1=a19 >
Smallest permutation representation of C40⋊6C4
►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)
(1 107 69 136)(2 86 70 155)(3 105 71 134)(4 84 72 153)(5 103 73 132)(6 82 74 151)(7 101 75 130)(8 120 76 149)(9 99 77 128)(10 118 78 147)(11 97 79 126)(12 116 80 145)(13 95 41 124)(14 114 42 143)(15 93 43 122)(16 112 44 141)(17 91 45 160)(18 110 46 139)(19 89 47 158)(20 108 48 137)(21 87 49 156)(22 106 50 135)(23 85 51 154)(24 104 52 133)(25 83 53 152)(26 102 54 131)(27 81 55 150)(28 100 56 129)(29 119 57 148)(30 98 58 127)(31 117 59 146)(32 96 60 125)(33 115 61 144)(34 94 62 123)(35 113 63 142)(36 92 64 121)(37 111 65 140)(38 90 66 159)(39 109 67 138)(40 88 68 157)
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), (1,107,69,136)(2,86,70,155)(3,105,71,134)(4,84,72,153)(5,103,73,132)(6,82,74,151)(7,101,75,130)(8,120,76,149)(9,99,77,128)(10,118,78,147)(11,97,79,126)(12,116,80,145)(13,95,41,124)(14,114,42,143)(15,93,43,122)(16,112,44,141)(17,91,45,160)(18,110,46,139)(19,89,47,158)(20,108,48,137)(21,87,49,156)(22,106,50,135)(23,85,51,154)(24,104,52,133)(25,83,53,152)(26,102,54,131)(27,81,55,150)(28,100,56,129)(29,119,57,148)(30,98,58,127)(31,117,59,146)(32,96,60,125)(33,115,61,144)(34,94,62,123)(35,113,63,142)(36,92,64,121)(37,111,65,140)(38,90,66,159)(39,109,67,138)(40,88,68,157)>;
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), (1,107,69,136)(2,86,70,155)(3,105,71,134)(4,84,72,153)(5,103,73,132)(6,82,74,151)(7,101,75,130)(8,120,76,149)(9,99,77,128)(10,118,78,147)(11,97,79,126)(12,116,80,145)(13,95,41,124)(14,114,42,143)(15,93,43,122)(16,112,44,141)(17,91,45,160)(18,110,46,139)(19,89,47,158)(20,108,48,137)(21,87,49,156)(22,106,50,135)(23,85,51,154)(24,104,52,133)(25,83,53,152)(26,102,54,131)(27,81,55,150)(28,100,56,129)(29,119,57,148)(30,98,58,127)(31,117,59,146)(32,96,60,125)(33,115,61,144)(34,94,62,123)(35,113,63,142)(36,92,64,121)(37,111,65,140)(38,90,66,159)(39,109,67,138)(40,88,68,157) );
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)], [(1,107,69,136),(2,86,70,155),(3,105,71,134),(4,84,72,153),(5,103,73,132),(6,82,74,151),(7,101,75,130),(8,120,76,149),(9,99,77,128),(10,118,78,147),(11,97,79,126),(12,116,80,145),(13,95,41,124),(14,114,42,143),(15,93,43,122),(16,112,44,141),(17,91,45,160),(18,110,46,139),(19,89,47,158),(20,108,48,137),(21,87,49,156),(22,106,50,135),(23,85,51,154),(24,104,52,133),(25,83,53,152),(26,102,54,131),(27,81,55,150),(28,100,56,129),(29,119,57,148),(30,98,58,127),(31,117,59,146),(32,96,60,125),(33,115,61,144),(34,94,62,123),(35,113,63,142),(36,92,64,121),(37,111,65,140),(38,90,66,159),(39,109,67,138),(40,88,68,157)]])
C40⋊6C4 is a maximal subgroup of
C40.6Q8 C40.Q8 D40⋊8C4 D8⋊2Dic5 C40⋊9Q8 C40.13Q8 C4×C40⋊C2 C8⋊Dic10 D40⋊9C4 Dic20⋊9C4 C23.34D20 C23.10D20 C23.38D20 C23.13D20 D4⋊Dic10 D4.Dic10 D10.16SD16 C40⋊6C4⋊C2 Q8⋊Dic10 Q8.2Dic10 D10.11SD16 D10⋊1C8.C2 Dic10.3Q8 D20⋊3Q8 D20.3Q8 Dic10⋊4Q8 C40⋊5Q8 C8.8Dic10 D5×C4.Q8 (C8×D5)⋊C4 C40⋊4Q8 C40⋊20(C2×C4) C23.22D20 C40⋊30D4 C23.47D20 C40⋊3D4 C40.4D4 D8⋊Dic5 C40⋊12D4 SD16×Dic5 C40⋊14D4 Q16⋊Dic5 C40.36D4 C60.Q8 C120⋊10C4
C40⋊6C4 is a maximal quotient of
C40⋊6C8 C40.Q8 C20.39C42 C60.Q8 C120⋊10C4
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 |
| type | + | + | + | - | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D5 | SD16 | Dic5 | D10 | Dic10 | D20 | C40⋊C2 |
| kernel | C40⋊6C4 | C4⋊Dic5 | C2×C40 | C40 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C40⋊6C4 ►in GL4(𝔽41) generated by
| 15 | 26 | 0 | 0 |
| 15 | 15 | 0 | 0 |
| 0 | 0 | 23 | 14 |
| 0 | 0 | 11 | 12 |
| 27 | 7 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [15,15,0,0,26,15,0,0,0,0,23,11,0,0,14,12],[27,7,0,0,7,14,0,0,0,0,1,0,0,0,40,40] >;
C40⋊6C4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_6C_4 % in TeX
G:=Group("C40:6C4"); // GroupNames label
G:=SmallGroup(160,24);
// by ID
G=gap.SmallGroup(160,24);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,579,69,4613]);
// Polycyclic
G:=Group<a,b|a^40=b^4=1,b*a*b^-1=a^19>;
// generators/relations
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