Copied to
clipboard

G = C2×C80order 160 = 25·5

Abelian group of type [2,80]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C80, SmallGroup(160,59)

Series: Derived Chief Lower central Upper central

C1 — C2×C80
C1C2C4C8C40C80 — C2×C80
C1 — C2×C80
C1 — C2×C80

Generators and relations for C2×C80
 G = < a,b | a2=b80=1, ab=ba >


Smallest permutation representation of C2×C80
Regular action on 160 points
Generators in S160
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)
(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)| (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144), (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( (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144), (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([(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,129),(66,130),(67,131),(68,132),(69,133),(70,134),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144)], [(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)])

C2×C80 is a maximal subgroup of
C80.9C4  C40.88D4  C8017C4  C40.78D4  C8013C4  C8014C4  C80.6C4  D101C16  D20.3C8  D407C4  D40.3C4  D20.6C8  D807C2

160 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D8A···8H10A···10L16A···16P20A···20P40A···40AF80A···80BL
order1222444455558···810···1016···1620···2040···4080···80
size1111111111111···11···11···11···11···11···1

160 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C4C5C8C8C10C10C16C20C20C40C40C80
kernelC2×C80C80C2×C40C40C2×C20C2×C16C20C2×C10C16C2×C8C10C8C2×C4C4C22C2
# reps12122444841688161664

Matrix representation of C2×C80 in GL2(𝔽241) generated by

2400
0240
,
2400
043
G:=sub<GL(2,GF(241))| [240,0,0,240],[240,0,0,43] >;

C2×C80 in GAP, Magma, Sage, TeX

C_2\times C_{80}
% in TeX

G:=Group("C2xC80");
// GroupNames label

G:=SmallGroup(160,59);
// by ID

G=gap.SmallGroup(160,59);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,69,88]);
// Polycyclic

G:=Group<a,b|a^2=b^80=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C80 in TeX

׿
×
𝔽