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## G = C23×C20order 160 = 25·5

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

Aliases: C23×C20, SmallGroup(160,228)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C20
 Chief series C1 — C2 — C10 — C20 — C2×C20 — C22×C20 — C23×C20
 Lower central C1 — C23×C20
 Upper central 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, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C24, C20, C2×C10, C23×C4, C2×C20, C22×C10, C22×C20, C23×C10, C23×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, C2×C20, C22×C10, C22×C20, C23×C10, C23×C20

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

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C23×C20 C22×C20 C23×C10 C22×C10 C23×C4 C22×C4 C24 C23 # reps 1 14 1 16 4 56 4 64

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

 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
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

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