C2^3xDic5 - GroupNames

G = C₂³×Dic₅ order 160 = 2⁵·5

Direct product of C₂³ and Dic₅

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂³×Dic₅, C₂⁴.₂D₅, C₁₀.₁₄C₂⁴, C₂³.₃₆D₁₀, C₅⋊₃(C₂³×C₄), C₁₀⋊₃(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

G = C23×Dic5 order 160 = 25·5

Direct product of C23 and Dic5

G = C₂³×Dic₅ order 160 = 2⁵·5

Direct product of C₂³ and Dic₅