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G = C4×C52C8order 160 = 25·5

Direct product of C4 and C52C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×C52C8, C204C8, C42.6D5, C10.6C42, C53(C4×C8), (C4×C20).6C2, C4.18(C4×D5), C20.44(C2×C4), (C2×C20).21C4, C10.15(C2×C8), (C2×C4).87D10, C2.1(C4×Dic5), (C2×C4).7Dic5, C22.6(C2×Dic5), (C2×C20).101C22, C2.1(C2×C52C8), (C2×C52C8).13C2, (C2×C10).44(C2×C4), SmallGroup(160,9)

Series: Derived Chief Lower central Upper central

C1C5 — C4×C52C8
C1C5C10C20C2×C20C2×C52C8 — C4×C52C8
C5 — C4×C52C8
C1C42

Generators and relations for C4×C52C8
 G = < a,b,c | a4=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

5C8
5C8
5C8
5C8
5C2×C8
5C2×C8
5C4×C8

Smallest permutation representation of C4×C52C8
Regular action on 160 points
Generators in S160
(1 85 61 69)(2 86 62 70)(3 87 63 71)(4 88 64 72)(5 81 57 65)(6 82 58 66)(7 83 59 67)(8 84 60 68)(9 91 20 136)(10 92 21 129)(11 93 22 130)(12 94 23 131)(13 95 24 132)(14 96 17 133)(15 89 18 134)(16 90 19 135)(25 73 121 149)(26 74 122 150)(27 75 123 151)(28 76 124 152)(29 77 125 145)(30 78 126 146)(31 79 127 147)(32 80 128 148)(33 45 100 141)(34 46 101 142)(35 47 102 143)(36 48 103 144)(37 41 104 137)(38 42 97 138)(39 43 98 139)(40 44 99 140)(49 116 110 158)(50 117 111 159)(51 118 112 160)(52 119 105 153)(53 120 106 154)(54 113 107 155)(55 114 108 156)(56 115 109 157)
(1 147 132 143 113)(2 114 144 133 148)(3 149 134 137 115)(4 116 138 135 150)(5 151 136 139 117)(6 118 140 129 152)(7 145 130 141 119)(8 120 142 131 146)(9 39 111 81 27)(10 28 82 112 40)(11 33 105 83 29)(12 30 84 106 34)(13 35 107 85 31)(14 32 86 108 36)(15 37 109 87 25)(16 26 88 110 38)(17 128 70 55 103)(18 104 56 71 121)(19 122 72 49 97)(20 98 50 65 123)(21 124 66 51 99)(22 100 52 67 125)(23 126 68 53 101)(24 102 54 69 127)(41 157 63 73 89)(42 90 74 64 158)(43 159 57 75 91)(44 92 76 58 160)(45 153 59 77 93)(46 94 78 60 154)(47 155 61 79 95)(48 96 80 62 156)
(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)| (1,85,61,69)(2,86,62,70)(3,87,63,71)(4,88,64,72)(5,81,57,65)(6,82,58,66)(7,83,59,67)(8,84,60,68)(9,91,20,136)(10,92,21,129)(11,93,22,130)(12,94,23,131)(13,95,24,132)(14,96,17,133)(15,89,18,134)(16,90,19,135)(25,73,121,149)(26,74,122,150)(27,75,123,151)(28,76,124,152)(29,77,125,145)(30,78,126,146)(31,79,127,147)(32,80,128,148)(33,45,100,141)(34,46,101,142)(35,47,102,143)(36,48,103,144)(37,41,104,137)(38,42,97,138)(39,43,98,139)(40,44,99,140)(49,116,110,158)(50,117,111,159)(51,118,112,160)(52,119,105,153)(53,120,106,154)(54,113,107,155)(55,114,108,156)(56,115,109,157), (1,147,132,143,113)(2,114,144,133,148)(3,149,134,137,115)(4,116,138,135,150)(5,151,136,139,117)(6,118,140,129,152)(7,145,130,141,119)(8,120,142,131,146)(9,39,111,81,27)(10,28,82,112,40)(11,33,105,83,29)(12,30,84,106,34)(13,35,107,85,31)(14,32,86,108,36)(15,37,109,87,25)(16,26,88,110,38)(17,128,70,55,103)(18,104,56,71,121)(19,122,72,49,97)(20,98,50,65,123)(21,124,66,51,99)(22,100,52,67,125)(23,126,68,53,101)(24,102,54,69,127)(41,157,63,73,89)(42,90,74,64,158)(43,159,57,75,91)(44,92,76,58,160)(45,153,59,77,93)(46,94,78,60,154)(47,155,61,79,95)(48,96,80,62,156), (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( (1,85,61,69)(2,86,62,70)(3,87,63,71)(4,88,64,72)(5,81,57,65)(6,82,58,66)(7,83,59,67)(8,84,60,68)(9,91,20,136)(10,92,21,129)(11,93,22,130)(12,94,23,131)(13,95,24,132)(14,96,17,133)(15,89,18,134)(16,90,19,135)(25,73,121,149)(26,74,122,150)(27,75,123,151)(28,76,124,152)(29,77,125,145)(30,78,126,146)(31,79,127,147)(32,80,128,148)(33,45,100,141)(34,46,101,142)(35,47,102,143)(36,48,103,144)(37,41,104,137)(38,42,97,138)(39,43,98,139)(40,44,99,140)(49,116,110,158)(50,117,111,159)(51,118,112,160)(52,119,105,153)(53,120,106,154)(54,113,107,155)(55,114,108,156)(56,115,109,157), (1,147,132,143,113)(2,114,144,133,148)(3,149,134,137,115)(4,116,138,135,150)(5,151,136,139,117)(6,118,140,129,152)(7,145,130,141,119)(8,120,142,131,146)(9,39,111,81,27)(10,28,82,112,40)(11,33,105,83,29)(12,30,84,106,34)(13,35,107,85,31)(14,32,86,108,36)(15,37,109,87,25)(16,26,88,110,38)(17,128,70,55,103)(18,104,56,71,121)(19,122,72,49,97)(20,98,50,65,123)(21,124,66,51,99)(22,100,52,67,125)(23,126,68,53,101)(24,102,54,69,127)(41,157,63,73,89)(42,90,74,64,158)(43,159,57,75,91)(44,92,76,58,160)(45,153,59,77,93)(46,94,78,60,154)(47,155,61,79,95)(48,96,80,62,156), (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([[(1,85,61,69),(2,86,62,70),(3,87,63,71),(4,88,64,72),(5,81,57,65),(6,82,58,66),(7,83,59,67),(8,84,60,68),(9,91,20,136),(10,92,21,129),(11,93,22,130),(12,94,23,131),(13,95,24,132),(14,96,17,133),(15,89,18,134),(16,90,19,135),(25,73,121,149),(26,74,122,150),(27,75,123,151),(28,76,124,152),(29,77,125,145),(30,78,126,146),(31,79,127,147),(32,80,128,148),(33,45,100,141),(34,46,101,142),(35,47,102,143),(36,48,103,144),(37,41,104,137),(38,42,97,138),(39,43,98,139),(40,44,99,140),(49,116,110,158),(50,117,111,159),(51,118,112,160),(52,119,105,153),(53,120,106,154),(54,113,107,155),(55,114,108,156),(56,115,109,157)], [(1,147,132,143,113),(2,114,144,133,148),(3,149,134,137,115),(4,116,138,135,150),(5,151,136,139,117),(6,118,140,129,152),(7,145,130,141,119),(8,120,142,131,146),(9,39,111,81,27),(10,28,82,112,40),(11,33,105,83,29),(12,30,84,106,34),(13,35,107,85,31),(14,32,86,108,36),(15,37,109,87,25),(16,26,88,110,38),(17,128,70,55,103),(18,104,56,71,121),(19,122,72,49,97),(20,98,50,65,123),(21,124,66,51,99),(22,100,52,67,125),(23,126,68,53,101),(24,102,54,69,127),(41,157,63,73,89),(42,90,74,64,158),(43,159,57,75,91),(44,92,76,58,160),(45,153,59,77,93),(46,94,78,60,154),(47,155,61,79,95),(48,96,80,62,156)], [(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)]])

C4×C52C8 is a maximal subgroup of
C42.279D10  C408C8  C20.53D8  C20.39SD16  D204C8  Dic104C8  C20.57D8  C20.26Q16  C20⋊C16  C42.4F5  C42.9F5  D5×C4×C8  C42.282D10  D10.7C42  C42.185D10  C42.196D10  Dic105C8  C42.198D10  D205C8  C206M4(2)  C42.6Dic5  C20.35C42  C42.187D10  C207M4(2)  C42.210D10  C42.213D10  C42.214D10  C42.215D10  C42.216D10  C20.16D8  C20⋊D8  C204SD16  C20.17D8  C20.SD16  C20.Q16  C206SD16  C20.D8  C203Q16  C20.11Q16
C4×C52C8 is a maximal quotient of
C408C8  C40.10C8  (C2×C20)⋊8C8

64 conjugacy classes

class 1 2A2B2C4A···4L5A5B8A···8P10A···10F20A···20X
order12224···4558···810···1020···20
size11111···1225···52···22···2

64 irreducible representations

dim11111122222
type++++-+
imageC1C2C2C4C4C8D5Dic5D10C52C8C4×D5
kernelC4×C52C8C2×C52C8C4×C20C52C8C2×C20C20C42C2×C4C2×C4C4C4
# reps1218416242168

Matrix representation of C4×C52C8 in GL3(𝔽41) generated by

100
0320
0032
,
100
0640
010
,
3800
010
0640
G:=sub<GL(3,GF(41))| [1,0,0,0,32,0,0,0,32],[1,0,0,0,6,1,0,40,0],[38,0,0,0,1,6,0,0,40] >;

C4×C52C8 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes_2C_8
% in TeX

G:=Group("C4xC5:2C8");
// GroupNames label

G:=SmallGroup(160,9);
// by ID

G=gap.SmallGroup(160,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,86,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×C52C8 in TeX

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