Copied to
clipboard

G = C23×C20order 160 = 25·5

Abelian group of type [2,2,2,20]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C20, SmallGroup(160,228)

Series: Derived Chief Lower central Upper central

C1 — C23×C20
C1C2C10C20C2×C20C22×C20 — C23×C20
C1 — C23×C20
C1 — C23×C20

Generators and relations for C23×C20
 G = < a,b,c,d | a2=b2=c2=d20=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 236, all normal (8 characteristic)
C1, C2, C2 [×14], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10, C10 [×14], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×C20 [×14], C23×C10, C23×C20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×C20 [×14], C23×C10, C23×C20

Smallest permutation representation of C23×C20
Regular action on 160 points
Generators in S160
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 141)(21 98)(22 99)(23 100)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 101)(79 102)(80 103)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 117)(42 118)(43 119)(44 120)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 156)(82 157)(83 158)(84 159)(85 160)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 61)(18 62)(19 63)(20 64)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 96)(42 97)(43 98)(44 99)(45 100)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 153)(120 154)
(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)| (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,141)(21,98)(22,99)(23,100)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,117)(42,118)(43,119)(44,120)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,61)(18,62)(19,63)(20,64)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154), (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( (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,141)(21,98)(22,99)(23,100)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,117)(42,118)(43,119)(44,120)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,61)(18,62)(19,63)(20,64)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154), (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([(1,142),(2,143),(3,144),(4,145),(5,146),(6,147),(7,148),(8,149),(9,150),(10,151),(11,152),(12,153),(13,154),(14,155),(15,156),(16,157),(17,158),(18,159),(19,160),(20,141),(21,98),(22,99),(23,100),(24,81),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,111),(69,112),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,101),(79,102),(80,103)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(41,117),(42,118),(43,119),(44,120),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133),(81,156),(82,157),(83,158),(84,159),(85,160),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,61),(18,62),(19,63),(20,64),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,96),(42,97),(43,98),(44,99),(45,100),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,141),(108,142),(109,143),(110,144),(111,145),(112,146),(113,147),(114,148),(115,149),(116,150),(117,151),(118,152),(119,153),(120,154)], [(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)])

C23×C20 is a maximal subgroup of   C24.4Dic5  C24.62D10  C24.63D10  C24.64D10  C24.65D10  C24.72D10

160 conjugacy classes

class 1 2A···2O4A···4P5A5B5C5D10A···10BH20A···20BL
order12···24···4555510···1020···20
size11···11···111111···11···1

160 irreducible representations

dim11111111
type+++
imageC1C2C2C4C5C10C10C20
kernelC23×C20C22×C20C23×C10C22×C10C23×C4C22×C4C24C23
# reps114116456464

Matrix representation of C23×C20 in GL4(𝔽41) generated by

40000
0100
00400
00040
,
1000
0100
00400
0001
,
1000
0100
0010
00040
,
32000
03200
00370
00020
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,37,0,0,0,0,20] >;

C23×C20 in GAP, Magma, Sage, TeX

C_2^3\times C_{20}
% in TeX

G:=Group("C2^3xC20");
// GroupNames label

G:=SmallGroup(160,228);
// by ID

G=gap.SmallGroup(160,228);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,480]);
// Polycyclic

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

׿
×
𝔽