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G = C2×C4×C20order 160 = 25·5

Abelian group of type [2,4,20]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C4×C20, SmallGroup(160,175)

Series: Derived Chief Lower central Upper central

C1 — C2×C4×C20
C1C2C22C2×C10C2×C20C4×C20 — C2×C4×C20
C1 — C2×C4×C20
C1 — C2×C4×C20

Generators and relations for C2×C4×C20
 G = < a,b,c | a2=b4=c20=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (8 characteristic)
C1, C2 [×7], C4 [×12], C22, C22 [×6], C5, C2×C4 [×18], C23, C10 [×7], C42 [×4], C22×C4 [×3], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C20 [×18], C22×C10, C4×C20 [×4], C22×C20 [×3], C2×C4×C20
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×18], C23, C10 [×7], C42 [×4], C22×C4 [×3], C20 [×12], C2×C10 [×7], C2×C42, C2×C20 [×18], C22×C10, C4×C20 [×4], C22×C20 [×3], C2×C4×C20

Smallest permutation representation of C2×C4×C20
Regular action on 160 points
Generators in S160
(1 38)(2 39)(3 40)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 121)(118 122)(119 123)(120 124)
(1 98 136 66)(2 99 137 67)(3 100 138 68)(4 81 139 69)(5 82 140 70)(6 83 121 71)(7 84 122 72)(8 85 123 73)(9 86 124 74)(10 87 125 75)(11 88 126 76)(12 89 127 77)(13 90 128 78)(14 91 129 79)(15 92 130 80)(16 93 131 61)(17 94 132 62)(18 95 133 63)(19 96 134 64)(20 97 135 65)(21 150 115 41)(22 151 116 42)(23 152 117 43)(24 153 118 44)(25 154 119 45)(26 155 120 46)(27 156 101 47)(28 157 102 48)(29 158 103 49)(30 159 104 50)(31 160 105 51)(32 141 106 52)(33 142 107 53)(34 143 108 54)(35 144 109 55)(36 145 110 56)(37 146 111 57)(38 147 112 58)(39 148 113 59)(40 149 114 60)
(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)| (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,121)(118,122)(119,123)(120,124), (1,98,136,66)(2,99,137,67)(3,100,138,68)(4,81,139,69)(5,82,140,70)(6,83,121,71)(7,84,122,72)(8,85,123,73)(9,86,124,74)(10,87,125,75)(11,88,126,76)(12,89,127,77)(13,90,128,78)(14,91,129,79)(15,92,130,80)(16,93,131,61)(17,94,132,62)(18,95,133,63)(19,96,134,64)(20,97,135,65)(21,150,115,41)(22,151,116,42)(23,152,117,43)(24,153,118,44)(25,154,119,45)(26,155,120,46)(27,156,101,47)(28,157,102,48)(29,158,103,49)(30,159,104,50)(31,160,105,51)(32,141,106,52)(33,142,107,53)(34,143,108,54)(35,144,109,55)(36,145,110,56)(37,146,111,57)(38,147,112,58)(39,148,113,59)(40,149,114,60), (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( (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,121)(118,122)(119,123)(120,124), (1,98,136,66)(2,99,137,67)(3,100,138,68)(4,81,139,69)(5,82,140,70)(6,83,121,71)(7,84,122,72)(8,85,123,73)(9,86,124,74)(10,87,125,75)(11,88,126,76)(12,89,127,77)(13,90,128,78)(14,91,129,79)(15,92,130,80)(16,93,131,61)(17,94,132,62)(18,95,133,63)(19,96,134,64)(20,97,135,65)(21,150,115,41)(22,151,116,42)(23,152,117,43)(24,153,118,44)(25,154,119,45)(26,155,120,46)(27,156,101,47)(28,157,102,48)(29,158,103,49)(30,159,104,50)(31,160,105,51)(32,141,106,52)(33,142,107,53)(34,143,108,54)(35,144,109,55)(36,145,110,56)(37,146,111,57)(38,147,112,58)(39,148,113,59)(40,149,114,60), (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([(1,38),(2,39),(3,40),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,125),(102,126),(103,127),(104,128),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,121),(118,122),(119,123),(120,124)], [(1,98,136,66),(2,99,137,67),(3,100,138,68),(4,81,139,69),(5,82,140,70),(6,83,121,71),(7,84,122,72),(8,85,123,73),(9,86,124,74),(10,87,125,75),(11,88,126,76),(12,89,127,77),(13,90,128,78),(14,91,129,79),(15,92,130,80),(16,93,131,61),(17,94,132,62),(18,95,133,63),(19,96,134,64),(20,97,135,65),(21,150,115,41),(22,151,116,42),(23,152,117,43),(24,153,118,44),(25,154,119,45),(26,155,120,46),(27,156,101,47),(28,157,102,48),(29,158,103,49),(30,159,104,50),(31,160,105,51),(32,141,106,52),(33,142,107,53),(34,143,108,54),(35,144,109,55),(36,145,110,56),(37,146,111,57),(38,147,112,58),(39,148,113,59),(40,149,114,60)], [(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×C4×C20 is a maximal subgroup of
C426Dic5  (C2×C20)⋊8C8  C2013M4(2)  C42.6Dic5  C42.7Dic5  C207(C4⋊C4)  (C2×C20)⋊10Q8  C424Dic5  C10.92(C4×D4)  C428Dic5  C429Dic5  C425Dic5  (C2×C4)⋊6D20  (C2×C42)⋊D5  C42.274D10  C42.276D10  C42.277D10

160 conjugacy classes

class 1 2A···2G4A···4X5A5B5C5D10A···10AB20A···20CR
order12···24···4555510···1020···20
size11···11···111111···11···1

160 irreducible representations

dim11111111
type+++
imageC1C2C2C4C5C10C10C20
kernelC2×C4×C20C4×C20C22×C20C2×C20C2×C42C42C22×C4C2×C4
# reps143244161296

Matrix representation of C2×C4×C20 in GL3(𝔽41) generated by

4000
010
0040
,
100
0400
009
,
3600
080
0040
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,9],[36,0,0,0,8,0,0,0,40] >;

C2×C4×C20 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{20}
% in TeX

G:=Group("C2xC4xC20");
// GroupNames label

G:=SmallGroup(160,175);
// by ID

G=gap.SmallGroup(160,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,487]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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