C5:C8:8D4 - GroupNames

G = C₅⋊C₈⋊₈D₄ order 320 = 2⁶·5

1^st semidirect product of C₅⋊C₈ and D₄ acting through Inn(C₅⋊C₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅⋊C₈⋊₈D₄, C₅⋊D₄⋊C₈, C₅⋊₁(C₈×D₄), Dic₅⋊(C₂×C₈), C₂.₁(D₄×F₅), D₁₀⋊₁(C₂×C₈), C₁₀.₁(C₄×D₄), C₂²⋊C₄.₈F₅, C₂²⋊₁(D₅⋊C₈), C₁₀.₂(C₈○D₄), C₁₀.₅(C₂²×C₈), D₁₀⋊C₈⋊₁₁C₂, C₂.₂(D₄.F₅), C₂³.₂₄(C₂×F₅), D₁₀⋊C₄.₅C₄, Dic₅.₆₅(C₂×D₄), C₁₀.D₄.₆C₄, C₂³.₂F₅⋊₃C₂, Dic₅⋊C₈⋊₁₂C₂, Dic₅.₅₀(C₄○D₄), Dic₅⋊₄D₄.₁₀C₂, C₂².₃₄(C₂²×F₅), (C₂×Dic₅).₃₂₂C₂³, (C₄×Dic₅).₂₄₇C₂², (C₂²×Dic₅).₁₇₇C₂², (C₄×C₅⋊C₈)⋊₁₁C₂, (C₂×C₁₀)⋊₁(C₂×C₈), (C₂×D₅⋊C₈)⋊₈C₂, (C₂²×C₅⋊C₈)⋊₁C₂, C₂.₇(C₂×D₅⋊C₈), (C₂×C₅⋊D₄).₃C₄, (C₂×C₄).₅₇(C₂×F₅), (C₂×C₂₀).₉₀(C₂×C₄), (C₅×C₂²⋊C₄).₈C₄, (C₂×C₅⋊C₈).₂₀C₂², (C₂×C₄×D₅).₂₈₆C₂², (C₂×C₁₀).₃₀(C₂²×C₄), (C₂²×C₁₀).₁₃(C₂×C₄), (C₂×Dic₅).₄₇(C₂×C₄), (C₂²×D₅).₃₉(C₂×C₄), SmallGroup(320,1030)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₅⋊C₈⋊₈D₄

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂×C₅⋊C₈ — C₂²×C₅⋊C₈ — C₅⋊C₈⋊₈D₄

Lower central

C₅ — C₁₀ — C₅⋊C₈⋊₈D₄

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₅⋊C₈⋊₈D₄
G = < a,b,c,d | a⁵=b⁸=c⁴=d²=1, bab^-1=a³, cac^-1=a^-1, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 442 in 134 conjugacy classes, 56 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×D₄, Dic₅, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₄×C₈, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂²×C₈, C₅⋊C₈, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, C₈×D₄, C₄×Dic₅, C₁₀.D₄, D₁₀⋊C₄, C₅×C₂²⋊C₄, D₅⋊C₈, C₂×C₅⋊C₈, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₄×C₅⋊C₈, D₁₀⋊C₈, Dic₅⋊C₈, C₂³.₂F₅, Dic₅⋊₄D₄, C₂×D₅⋊C₈, C₂²×C₅⋊C₈, C₅⋊C₈⋊₈D₄
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂³, C₂×C₈, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂²×C₈, C₈○D₄, C₂×F₅, C₈×D₄, D₅⋊C₈, C₂²×F₅, C₂×D₅⋊C₈, D₄.F₅, D₄×F₅, C₅⋊C₈⋊₈D₄

Smallest permutation representation of C₅⋊C₈⋊₈D₄
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	5	8A	···	8H	8I	···	8T	10A	10B	10C	10D	10E	20A	20B	20C	20D
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	5	8	···	8	8	···	8	10	10	10	10	10	20	20	20	20
size	1	1	1	1	2	2	10	10	2	2	2	2	5	5	5	5	10	10	10	10	4	5	···	5	10	···	10	4	4	4	8	8	8	8	8	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	8	8
type	+	+	+	+	+	+	+	+						+			+	+	+		-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈	D₄	C₄○D₄	C₈○D₄	F₅	C₂×F₅	C₂×F₅	D₅⋊C₈	D₄.F₅	D₄×F₅
kernel	C₅⋊C₈⋊₈D₄	C₄×C₅⋊C₈	D₁₀⋊C₈	Dic₅⋊C₈	C₂³.₂F₅	Dic₅⋊₄D₄	C₂×D₅⋊C₈	C₂²×C₅⋊C₈	C₁₀.D₄	D₁₀⋊C₄	C₅×C₂²⋊C₄	C₂×C₅⋊D₄	C₅⋊D₄	C₅⋊C₈	Dic₅	C₁₀	C₂²⋊C₄	C₂×C₄	C₂³	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	2	2	2	16	2	2	4	1	2	1	4	1	1

Matrix representation of C₅⋊C₈⋊₈D₄ ►in GL₆(𝔽₄₁)

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

9	0	0	0	0	0
0	9	0	0	0	0
0	0	19	0	22	21
0	0	0	21	1	40
0	0	20	40	1	21
0	0	20	21	22	0

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

0	1	0	0	0	0
1	0	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

G:=sub<GL(6,GF(41))| [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],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,19,0,20,20,0,0,0,21,40,21,0,0,22,1,1,22,0,0,21,40,21,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40,0,0,0,0,40,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C₅⋊C₈⋊₈D₄ in GAP, Magma, Sage, TeX

C_5\rtimes C_8\rtimes_8D_4

% in TeX

G:=Group("C5:C8:8D4");

// GroupNames label

G:=SmallGroup(320,1030);

// by ID

G=gap.SmallGroup(320,1030);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^3,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = C5⋊C8⋊8D4 order 320 = 26·5

1st semidirect product of C5⋊C8 and D4 acting through Inn(C5⋊C8)

G = C₅⋊C₈⋊₈D₄ order 320 = 2⁶·5

1^st semidirect product of C₅⋊C₈ and D₄ acting through Inn(C₅⋊C₈)