C2^2xDic10 - GroupNames

G = C₂²×Dic₁₀ order 160 = 2⁵·5

Direct product of C₂² and Dic₁₀

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

Aliases: C₂²×Dic₁₀, C₁₀.₁C₂⁴, C₂₀.₃₄C₂³, C₂³.₃₃D₁₀, Dic₅.₁C₂³, (C₂×C₁₀)⋊₄Q₈, C₁₀⋊₁(C₂×Q₈), C₅⋊₁(C₂²×Q₈), (C₂×C₄).₈₆D₁₀, (C₂²×C₄).₈D₅, C₂.₃(C₂³×D₅), (C₂²×C₂₀).₈C₂, C₄.₃₂(C₂²×D₅), (C₂×C₂₀).₉₅C₂², (C₂×C₁₀).₆₂C₂³, (C₂²×Dic₅).₆C₂, C₂².₂₈(C₂²×D₅), (C₂²×C₁₀).₄₃C₂², (C₂×Dic₅).₄₅C₂², SmallGroup(160,213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂²×Dic₁₀

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂²×Dic₅ — C₂²×Dic₁₀

Lower central

C₅ — C₁₀ — C₂²×Dic₁₀

Upper central

C₁ — C₂³ — C₂²×C₄

Generators and relations for C₂²×Dic₁₀
G = < a,b,c,d | a²=b²=c²⁰=1, d²=c¹⁰, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 360 in 156 conjugacy classes, 105 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₀, C₁₀, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₅, C₂₀, C₂×C₁₀, C₂²×Q₈, Dic₁₀, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂²×Dic₁₀
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, C₂⁴, D₁₀, C₂²×Q₈, Dic₁₀, C₂²×D₅, C₂×Dic₁₀, C₂³×D₅, C₂²×Dic₁₀

Smallest permutation representation of C₂²×Dic₁₀
►Regular action on 160 points

Generators in S₁₆₀

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

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

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

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

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

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

C₂²×Dic₁₀ is a maximal subgroup of
(C₂×C₂₀)⋊Q₈ (C₂×Dic₅)⋊Q₈ (C₂×C₄).₂₀D₂₀ Dic₁₀⋊₁₄D₄ C₂²⋊Dic₂₀ (C₂×C₂₀)⋊₁₀Q₈ C₂³⋊Dic₁₀ (C₂×Dic₅)⋊₆Q₈ (C₂×C₄)⋊Dic₁₀ C₄.(C₂×D₂₀) Dic₁₀⋊₁₇D₄ Dic₁₀.₃₇D₄ C₂³.₄₆D₂₀ C₄².₈₇D₁₀ C₄².₉₂D₁₀ Dic₁₀⋊₂₃D₄ Dic₁₀⋊₁₉D₄ Dic₁₀⋊₂₁D₄ C₁₀.₇₉2_-¹⁺⁴ C₁₀.₁₀₅2_-¹⁺⁴ C₂²×Q₈×D₅
C₂²×Dic₁₀ is a maximal quotient of
C₄².₂₇₄D₁₀ C₂³⋊₂Dic₁₀ C₁₀.₁2_-¹⁺⁴ C₄².₈₈D₁₀ C₄².₉₀D₁₀ D₄⋊₅Dic₁₀ D₄⋊₆Dic₁₀ Q₈⋊₅Dic₁₀ Q₈⋊₆Dic₁₀

52 conjugacy classes

class	1	2A	···	2G	4A	4B	4C	4D	4E	···	4L	5A	5B	10A	···	10N	20A	···	20P
order	1	2	···	2	4	4	4	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	2	2	2	2	10	···	10	2	2	2	···	2	2	···	2

52 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+	+	-	+	+	+	-
image	C₁	C₂	C₂	C₂	Q₈	D₅	D₁₀	D₁₀	Dic₁₀
kernel	C₂²×Dic₁₀	C₂×Dic₁₀	C₂²×Dic₅	C₂²×C₂₀	C₂×C₁₀	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	12	2	1	4	2	12	2	16

Matrix representation of C₂²×Dic₁₀ ►in GL₄(𝔽₄₁) generated by

40	0	0	0
0	40	0	0
0	0	1	0
0	0	0	1

40	0	0	0
0	1	0	0
0	0	40	0
0	0	0	40

40	0	0	0
0	1	0	0
0	0	28	39
0	0	2	16

40	0	0	0
0	40	0	0
0	0	28	9
0	0	13	13

G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,28,2,0,0,39,16],[40,0,0,0,0,40,0,0,0,0,28,13,0,0,9,13] >;

C₂²×Dic₁₀ in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{10}

% in TeX

G:=Group("C2^2xDic10");

// GroupNames label

G:=SmallGroup(160,213);

// by ID

G=gap.SmallGroup(160,213);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,579,69,4613]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = C22×Dic10 order 160 = 25·5

Direct product of C22 and Dic10

G = C₂²×Dic₁₀ order 160 = 2⁵·5

Direct product of C₂² and Dic₁₀