direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊C8, C4⋊C40, C20⋊5C8, C20.67D4, C20.12Q8, C42.2C10, C10.14M4(2), C4.4(C5×Q8), C2.2(C2×C40), (C4×C20).8C2, (C2×C40).4C2, (C2×C8).2C10, (C2×C4).4C20, C4.18(C5×D4), (C2×C20).24C4, C10.21(C2×C8), C10.18(C4⋊C4), C2.3(C5×M4(2)), C22.10(C2×C20), (C2×C20).136C22, C2.2(C5×C4⋊C4), (C2×C4).32(C2×C10), (C2×C10).59(C2×C4), SmallGroup(160,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊C8
G = < a,b,c | a5=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C4⋊C8
►Regular action on 160 points
Generators in S160
(1 117 37 109 29)(2 118 38 110 30)(3 119 39 111 31)(4 120 40 112 32)(5 113 33 105 25)(6 114 34 106 26)(7 115 35 107 27)(8 116 36 108 28)(9 20 43 97 95)(10 21 44 98 96)(11 22 45 99 89)(12 23 46 100 90)(13 24 47 101 91)(14 17 48 102 92)(15 18 41 103 93)(16 19 42 104 94)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 67 127 41)(2 42 128 68)(3 69 121 43)(4 44 122 70)(5 71 123 45)(6 46 124 72)(7 65 125 47)(8 48 126 66)(9 111 77 129)(10 130 78 112)(11 105 79 131)(12 132 80 106)(13 107 73 133)(14 134 74 108)(15 109 75 135)(16 136 76 110)(17 54 146 28)(18 29 147 55)(19 56 148 30)(20 31 149 49)(21 50 150 32)(22 25 151 51)(23 52 152 26)(24 27 145 53)(33 159 59 89)(34 90 60 160)(35 153 61 91)(36 92 62 154)(37 155 63 93)(38 94 64 156)(39 157 57 95)(40 96 58 158)(81 137 97 119)(82 120 98 138)(83 139 99 113)(84 114 100 140)(85 141 101 115)(86 116 102 142)(87 143 103 117)(88 118 104 144)
(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)
G:=sub<Sym(160)| (1,117,37,109,29)(2,118,38,110,30)(3,119,39,111,31)(4,120,40,112,32)(5,113,33,105,25)(6,114,34,106,26)(7,115,35,107,27)(8,116,36,108,28)(9,20,43,97,95)(10,21,44,98,96)(11,22,45,99,89)(12,23,46,100,90)(13,24,47,101,91)(14,17,48,102,92)(15,18,41,103,93)(16,19,42,104,94)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,67,127,41)(2,42,128,68)(3,69,121,43)(4,44,122,70)(5,71,123,45)(6,46,124,72)(7,65,125,47)(8,48,126,66)(9,111,77,129)(10,130,78,112)(11,105,79,131)(12,132,80,106)(13,107,73,133)(14,134,74,108)(15,109,75,135)(16,136,76,110)(17,54,146,28)(18,29,147,55)(19,56,148,30)(20,31,149,49)(21,50,150,32)(22,25,151,51)(23,52,152,26)(24,27,145,53)(33,159,59,89)(34,90,60,160)(35,153,61,91)(36,92,62,154)(37,155,63,93)(38,94,64,156)(39,157,57,95)(40,96,58,158)(81,137,97,119)(82,120,98,138)(83,139,99,113)(84,114,100,140)(85,141,101,115)(86,116,102,142)(87,143,103,117)(88,118,104,144), (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)>;
G:=Group( (1,117,37,109,29)(2,118,38,110,30)(3,119,39,111,31)(4,120,40,112,32)(5,113,33,105,25)(6,114,34,106,26)(7,115,35,107,27)(8,116,36,108,28)(9,20,43,97,95)(10,21,44,98,96)(11,22,45,99,89)(12,23,46,100,90)(13,24,47,101,91)(14,17,48,102,92)(15,18,41,103,93)(16,19,42,104,94)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,67,127,41)(2,42,128,68)(3,69,121,43)(4,44,122,70)(5,71,123,45)(6,46,124,72)(7,65,125,47)(8,48,126,66)(9,111,77,129)(10,130,78,112)(11,105,79,131)(12,132,80,106)(13,107,73,133)(14,134,74,108)(15,109,75,135)(16,136,76,110)(17,54,146,28)(18,29,147,55)(19,56,148,30)(20,31,149,49)(21,50,150,32)(22,25,151,51)(23,52,152,26)(24,27,145,53)(33,159,59,89)(34,90,60,160)(35,153,61,91)(36,92,62,154)(37,155,63,93)(38,94,64,156)(39,157,57,95)(40,96,58,158)(81,137,97,119)(82,120,98,138)(83,139,99,113)(84,114,100,140)(85,141,101,115)(86,116,102,142)(87,143,103,117)(88,118,104,144), (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) );
G=PermutationGroup([[(1,117,37,109,29),(2,118,38,110,30),(3,119,39,111,31),(4,120,40,112,32),(5,113,33,105,25),(6,114,34,106,26),(7,115,35,107,27),(8,116,36,108,28),(9,20,43,97,95),(10,21,44,98,96),(11,22,45,99,89),(12,23,46,100,90),(13,24,47,101,91),(14,17,48,102,92),(15,18,41,103,93),(16,19,42,104,94),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,85,153,73,145),(66,86,154,74,146),(67,87,155,75,147),(68,88,156,76,148),(69,81,157,77,149),(70,82,158,78,150),(71,83,159,79,151),(72,84,160,80,152)], [(1,67,127,41),(2,42,128,68),(3,69,121,43),(4,44,122,70),(5,71,123,45),(6,46,124,72),(7,65,125,47),(8,48,126,66),(9,111,77,129),(10,130,78,112),(11,105,79,131),(12,132,80,106),(13,107,73,133),(14,134,74,108),(15,109,75,135),(16,136,76,110),(17,54,146,28),(18,29,147,55),(19,56,148,30),(20,31,149,49),(21,50,150,32),(22,25,151,51),(23,52,152,26),(24,27,145,53),(33,159,59,89),(34,90,60,160),(35,153,61,91),(36,92,62,154),(37,155,63,93),(38,94,64,156),(39,157,57,95),(40,96,58,158),(81,137,97,119),(82,120,98,138),(83,139,99,113),(84,114,100,140),(85,141,101,115),(86,116,102,142),(87,143,103,117),(88,118,104,144)], [(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)]])
C5×C4⋊C8 is a maximal subgroup of
C20.53D8 C20.39SD16 C4.Dic20 C20.47D8 D20⋊4C8 Dic10⋊4C8 C4.D40 C20.2D8 Dic5.5M4(2) Dic10.3Q8 Dic10⋊5C8 C42.198D10 C42.200D10 D20⋊5C8 C42.202D10 D10⋊5M4(2) C20⋊5M4(2) C20⋊6M4(2) C42.30D10 C42.31D10 C20⋊SD16 D20⋊3Q8 C4⋊D40 D20.19D4 C42.36D10 D20⋊4Q8 D20.3Q8 Dic10⋊8D4 C4⋊Dic20 C20.7Q16 Dic10⋊4Q8 D4×C40 Q8×C40
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C8 | C10 | C10 | C20 | C40 | D4 | Q8 | M4(2) | C5×D4 | C5×Q8 | C5×M4(2) |
| kernel | C5×C4⋊C8 | C4×C20 | C2×C40 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C2×C4 | C4 | C20 | C20 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 32 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×C4⋊C8 ►in GL3(𝔽41) generated by
| 10 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 38 | 0 | 0 |
| 0 | 28 | 23 |
| 0 | 23 | 13 |
G:=sub<GL(3,GF(41))| [10,0,0,0,18,0,0,0,18],[1,0,0,0,0,40,0,1,0],[38,0,0,0,28,23,0,23,13] >;
C5×C4⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes C_8
% in TeX
G:=Group("C5xC4:C8"); // GroupNames label
G:=SmallGroup(160,55);
// by ID
G=gap.SmallGroup(160,55);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,127,88]);
// Polycyclic
G:=Group<a,b,c|a^5=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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