C10xC4^2:2C2 - GroupNames

G = C₁₀×C₄²⋊₂C₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₄²⋊₂C₂

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C₁₀×C₄²⋊₂C₂, (C₂×C₄²)⋊₄C₁₀, (C₄×C₂₀)⋊₄₉C₂², C₄²⋊₁₄(C₂×C₁₀), C₂⁴.₁₄(C₂×C₁₀), (C₂×C₂₀).₇₁₂C₂³, (C₂×C₁₀).₃₄₈C₂⁴, (C₂³×C₁₀).₁₄C₂², C₂².₂₂(C₂³×C₁₀), C₂³.₇₂(C₂²×C₁₀), (C₂²×C₁₀).₈₆C₂³, (C₂²×C₂₀).₅₁₀C₂², (C₂×C₄×C₂₀)⋊₆C₂, (C₂×C₄⋊C₄)⋊₁₇C₁₀, (C₁₀×C₄⋊C₄)⋊₄₄C₂, C₄⋊C₄⋊₁₃(C₂×C₁₀), (C₅×C₄⋊C₄)⋊₆₉C₂², C₂.₁₁(C₁₀×C₄○D₄), C₁₀.₂₃₀(C₂×C₄○D₄), C₂².₃₄(C₅×C₄○D₄), (C₁₀×C₂²⋊C₄).₃₂C₂, C₂²⋊C₄.₁₁(C₂×C₁₀), (C₂×C₂²⋊C₄).₁₂C₁₀, (C₂×C₄).₁₆(C₂²×C₁₀), (C₂×C₁₀).₂₃₄(C₄○D₄), (C₂²×C₄).₁₀₂(C₂×C₁₀), (C₅×C₂²⋊C₄).₁₄₅C₂², SmallGroup(320,1530)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₁₀×C₄²⋊₂C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂²×C₁₀ — C₅×C₂²⋊C₄ — C₅×C₄²⋊₂C₂ — C₁₀×C₄²⋊₂C₂

Lower central

C₁ — C₂² — C₁₀×C₄²⋊₂C₂

Upper central

C₁ — C₂²×C₁₀ — C₁₀×C₄²⋊₂C₂

Generators and relations for C₁₀×C₄²⋊₂C₂
G = < a,b,c,d | a¹⁰=b⁴=c⁴=d²=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc², dcd=b²c^-1 >

Subgroups: 354 in 246 conjugacy classes, 162 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₄²⋊₂C₂, C₄×C₂₀, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂²×C₂₀, C₂³×C₁₀, C₂×C₄×C₂₀, C₁₀×C₂²⋊C₄, C₁₀×C₄⋊C₄, C₅×C₄²⋊₂C₂, C₁₀×C₄²⋊₂C₂
Quotients: C₁, C₂, C₂², C₅, C₂³, C₁₀, C₄○D₄, C₂⁴, C₂×C₁₀, C₄²⋊₂C₂, C₂×C₄○D₄, C₂²×C₁₀, C₂×C₄²⋊₂C₂, C₅×C₄○D₄, C₂³×C₁₀, C₅×C₄²⋊₂C₂, C₁₀×C₄○D₄, C₁₀×C₄²⋊₂C₂

Smallest permutation representation of C₁₀×C₄²⋊₂C₂
►On 160 points

Generators in S₁₆₀

(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)

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

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

(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 151)(20 152)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)

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

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

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

140 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4L	4M	···	4R	5A	5B	5C	5D	10A	···	10AB	10AC	···	10AJ	20A	···	20AV	20AW	···	20BT
order	1	2	···	2	2	2	4	···	4	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20	20	···	20
size	1	1	···	1	4	4	2	···	2	4	···	4	1	1	1	1	1	···	1	4	···	4	2	···	2	4	···	4

140 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₄○D₄	C₅×C₄○D₄
kernel	C₁₀×C₄²⋊₂C₂	C₂×C₄×C₂₀	C₁₀×C₂²⋊C₄	C₁₀×C₄⋊C₄	C₅×C₄²⋊₂C₂	C₂×C₄²⋊₂C₂	C₂×C₄²	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₄²⋊₂C₂	C₂×C₁₀	C₂²
# reps	1	1	3	3	8	4	4	12	12	32	12	48

Matrix representation of C₁₀×C₄²⋊₂C₂ ►in GL₅(𝔽₄₁)

40	0	0	0	0
0	23	0	0	0
0	0	23	0	0
0	0	0	18	0
0	0	0	0	18

40	0	0	0	0
0	20	39	0	0
0	16	21	0	0
0	0	0	32	0
0	0	0	0	32

40	0	0	0	0
0	32	0	0	0
0	0	32	0	0
0	0	0	0	40
0	0	0	40	0

40	0	0	0	0
0	1	0	0	0
0	20	40	0	0
0	0	0	1	0
0	0	0	0	40

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,18,0,0,0,0,0,18],[40,0,0,0,0,0,20,16,0,0,0,39,21,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,40,0],[40,0,0,0,0,0,1,20,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;

C₁₀×C₄²⋊₂C₂ in GAP, Magma, Sage, TeX

C_{10}\times C_4^2\rtimes_2C_2

% in TeX

G:=Group("C10xC4^2:2C2");

// GroupNames label

G:=SmallGroup(320,1530);

// by ID

G=gap.SmallGroup(320,1530);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,436]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;

// generators/relations

G = C10×C42⋊2C2 order 320 = 26·5

Direct product of C10 and C42⋊2C2

G = C₁₀×C₄²⋊₂C₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₄²⋊₂C₂