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## G = C23×Dic5order 160 = 25·5

### Direct product of C23 and Dic5

Aliases: C23×Dic5, C24.2D5, C10.14C24, C23.36D10, C53(C23×C4), C103(C22×C4), (C22×C10)⋊8C4, C2.2(C23×D5), (C23×C10).3C2, (C2×C10).69C23, C22.33(C22×D5), (C22×C10).47C22, (C2×C10)⋊12(C2×C4), SmallGroup(160,226)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C23×Dic5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22×Dic5 — C23×Dic5
 Lower central C5 — C23×Dic5
 Upper central C1 — C24

Generators and relations for C23×Dic5
G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 440 in 236 conjugacy classes, 185 normal (7 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C24, Dic5, C2×C10, C23×C4, C2×Dic5, C22×C10, C22×Dic5, C23×C10, C23×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, C2×Dic5, C22×D5, C22×Dic5, C23×D5, C23×Dic5

Smallest permutation representation of C23×Dic5
Regular action on 160 points
Generators in S160
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 71)(9 72)(10 73)(11 66)(12 67)(13 68)(14 69)(15 70)(16 61)(17 62)(18 63)(19 64)(20 65)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)(81 156)(82 157)(83 158)(84 159)(85 160)(86 151)(87 152)(88 153)(89 154)(90 155)(91 146)(92 147)(93 148)(94 149)(95 150)(96 141)(97 142)(98 143)(99 144)(100 145)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(111 126)(112 127)(113 128)(114 129)(115 130)(116 121)(117 122)(118 123)(119 124)(120 125)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 31)(9 32)(10 33)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)
(1 19)(2 20)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(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)
(1 106 6 101)(2 105 7 110)(3 104 8 109)(4 103 9 108)(5 102 10 107)(11 114 16 119)(12 113 17 118)(13 112 18 117)(14 111 19 116)(15 120 20 115)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 144 46 149)(42 143 47 148)(43 142 48 147)(44 141 49 146)(45 150 50 145)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(61 124 66 129)(62 123 67 128)(63 122 68 127)(64 121 69 126)(65 130 70 125)(71 134 76 139)(72 133 77 138)(73 132 78 137)(74 131 79 136)(75 140 80 135)

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

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

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

C23×Dic5 is a maximal subgroup of
C24.44D10  C23.42D20  C24.46D10  C24.47D10  C23.45D20  C24.18D10  C24.4F5  C24.56D10  D5×C23×C4
C23×Dic5 is a maximal quotient of
C24.38D10  C10.422- 1+4  C20.76C24  C10.1062- 1+4

64 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 5A 5B 10A ··· 10AD order 1 2 ··· 2 4 ··· 4 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D5 Dic5 D10 kernel C23×Dic5 C22×Dic5 C23×C10 C22×C10 C24 C23 C23 # reps 1 14 1 16 2 16 14

Matrix representation of C23×Dic5 in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1 40 0 0 0 36 6
,
 1 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 0 19 0 0 0 28 0

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,36,0,0,0,40,6],[1,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,19,0] >;

C23×Dic5 in GAP, Magma, Sage, TeX

C_2^3\times {\rm Dic}_5
% in TeX

G:=Group("C2^3xDic5");
// GroupNames label

G:=SmallGroup(160,226);
// by ID

G=gap.SmallGroup(160,226);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽