C2^2xC5:C8 - GroupNames

G = C₂²×C₅⋊C₈ order 160 = 2⁵·5

Direct product of C₂² and C₅⋊C₈

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

Aliases: C₂²×C₅⋊C₈, C₂³.₄F₅, Dic₅.₁₄C₂³, (C₂×C₁₀)⋊₂C₈, C₁₀⋊₂(C₂×C₈), C₅⋊₂(C₂²×C₈), C₂.₃(C₂²×F₅), (C₂²×C₁₀).₅C₄, C₂².₁₉(C₂×F₅), C₁₀.₁₁(C₂²×C₄), (C₂×Dic₅).₁₃C₄, Dic₅.₁₈(C₂×C₄), (C₂²×Dic₅).₈C₂, (C₂×Dic₅).₅₇C₂², (C₂×C₁₀).₁₉(C₂×C₄), SmallGroup(160,210)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂²×C₅⋊C₈

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂²×C₅⋊C₈

Lower central

C₅ — C₂²×C₅⋊C₈

Upper central

C₁ — C₂³

Generators and relations for C₂²×C₅⋊C₈
G = < a,b,c,d | a²=b²=c⁵=d⁸=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 164 in 76 conjugacy classes, 54 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₈, C₂×C₄, C₂³, C₁₀, C₁₀, C₂×C₈, C₂²×C₄, Dic₅, Dic₅, C₂×C₁₀, C₂²×C₈, C₅⋊C₈, C₂×Dic₅, C₂²×C₁₀, C₂×C₅⋊C₈, C₂²×Dic₅, C₂²×C₅⋊C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, F₅, C₂²×C₈, C₅⋊C₈, C₂×F₅, C₂×C₅⋊C₈, C₂²×F₅, C₂²×C₅⋊C₈

Smallest permutation representation of C₂²×C₅⋊C₈
►Regular action on 160 points

Generators in S₁₆₀

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

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

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

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

C₂²×C₅⋊C₈ is a maximal subgroup of C₁₀.(C₄⋊C₈) Dic₅.C₄² C₅⋊C₈⋊₈D₄ C₅⋊C₈⋊D₄ C₂₀⋊C₈⋊C₂ C₅⋊C₈⋊₇D₄ (C₂×D₄).₇F₅
C₂²×C₅⋊C₈ is a maximal quotient of Dic₅.₁₂M₄(2) C₅⋊C₁₆.C₂²

40 conjugacy classes

class	1	2A	···	2G	4A	···	4H	5	8A	···	8P	10A	···	10G
order	1	2	···	2	4	···	4	5	8	···	8	10	···	10
size	1	1	···	1	5	···	5	4	5	···	5	4	···	4

40 irreducible representations

dim	1	1	1	1	1	1	4	4	4
type	+	+	+				+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	F₅	C₅⋊C₈	C₂×F₅
kernel	C₂²×C₅⋊C₈	C₂×C₅⋊C₈	C₂²×Dic₅	C₂×Dic₅	C₂²×C₁₀	C₂×C₁₀	C₂³	C₂²	C₂²
# reps	1	6	1	6	2	16	1	4	3

Matrix representation of C₂²×C₅⋊C₈ ►in GL₆(𝔽₄₁)

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	30	24	6	39
0	0	36	22	30	28
0	0	19	11	13	34
0	0	2	17	11	17

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,36,19,2,0,0,24,22,11,17,0,0,6,30,13,11,0,0,39,28,34,17] >;

C₂²×C₅⋊C₈ in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes C_8

% in TeX

G:=Group("C2^2xC5:C8");

// GroupNames label

G:=SmallGroup(160,210);

// by ID

G=gap.SmallGroup(160,210);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,69,2309,599]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^5=d^8=1,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^3>;

// generators/relations

G = C22×C5⋊C8 order 160 = 25·5

Direct product of C22 and C5⋊C8

G = C₂²×C₅⋊C₈ order 160 = 2⁵·5

Direct product of C₂² and C₅⋊C₈