C5:C8:7D4 - GroupNames

G = C₅⋊C₈⋊₇D₄ order 320 = 2⁶·5

4^th semidirect product of C₅⋊C₈ and D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅⋊C₈⋊₇D₄, C₅⋊₄(C₈⋊₉D₄), (C₂×D₄).₅F₅, C₂.₂₇(D₄×F₅), (D₄×C₁₀).₄C₄, C₁₀.₂₇(C₄×D₄), C₄⋊Dic₅.₇C₄, (C₂×C₁₀)⋊₂M₄(2), C₂³.D₅.₆C₄, C₂³.₁₁(C₂×F₅), Dic₅⋊C₈⋊₃C₂, C₁₀.₁₅(C₈○D₄), Dic₅.₇₉(C₂×D₄), (D₄×Dic₅).₁₀C₂, C₂.₁₅(D₄.F₅), C₂³.₂F₅⋊₉C₂, C₁₀.C₄²⋊₄C₂, C₁₀.₃₀(C₂×M₄(2)), C₂²⋊₁(C₂².F₅), Dic₅.₅₈(C₄○D₄), C₂².₉₁(C₂²×F₅), (C₄×Dic₅).₆₉C₂², (C₂×Dic₅).₃₅₂C₂³, (C₂²×Dic₅).₁₈₅C₂², (C₂²×C₅⋊C₈)⋊₅C₂, (C₂×C₄).₃₆(C₂×F₅), (C₂×C₂₀).₂₃(C₂×C₄), (C₂×C₅⋊C₈).₄₀C₂², (C₂×C₂².F₅)⋊₄C₂, C₂.₉(C₂×C₂².F₅), (C₂²×C₁₀).₂₄(C₂×C₄), (C₂×C₁₀).₇₆(C₂²×C₄), (C₂×Dic₅).₇₁(C₂×C₄), SmallGroup(320,1111)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₅⋊C₈⋊₇D₄

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂×C₅⋊C₈ — C₂²×C₅⋊C₈ — C₅⋊C₈⋊₇D₄

Lower central

C₅ — C₂×C₁₀ — C₅⋊C₈⋊₇D₄

Upper central

C₁ — C₂² — C₂×D₄

Generators and relations for C₅⋊C₈⋊₇D₄
G = < a,b,c,d | a⁵=b⁸=c⁴=d²=1, bab^-1=a³, ac=ca, ad=da, cbc^-1=b⁵, bd=db, dcd=c^-1 >

Subgroups: 394 in 124 conjugacy classes, 48 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, Dic₅, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂²×C₈, C₂×M₄(2), C₅⋊C₈, C₅⋊C₈, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₈⋊₉D₄, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂×C₅⋊C₈, C₂×C₅⋊C₈, C₂².F₅, C₂²×Dic₅, D₄×C₁₀, C₁₀.C₄², Dic₅⋊C₈, C₂³.₂F₅, D₄×Dic₅, C₂²×C₅⋊C₈, C₂×C₂².F₅, C₅⋊C₈⋊₇D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×M₄(2), C₈○D₄, C₂×F₅, C₈⋊₉D₄, C₂².F₅, C₂²×F₅, D₄.F₅, D₄×F₅, C₂×C₂².F₅, C₅⋊C₈⋊₇D₄

Smallest permutation representation of C₅⋊C₈⋊₇D₄
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	8	8
type	+	+	+	+	+	+	+				+				+	+	+	-	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	M₄(2)	C₈○D₄	F₅	C₂×F₅	C₂×F₅	C₂².F₅	D₄.F₅	D₄×F₅
kernel	C₅⋊C₈⋊₇D₄	C₁₀.C₄²	Dic₅⋊C₈	C₂³.₂F₅	D₄×Dic₅	C₂²×C₅⋊C₈	C₂×C₂².F₅	C₄⋊Dic₅	C₂³.D₅	D₄×C₁₀	C₅⋊C₈	Dic₅	C₂×C₁₀	C₁₀	C₂×D₄	C₂×C₄	C₂³	C₂²	C₂	C₂
# reps	1	1	1	2	1	1	1	2	4	2	2	2	4	4	1	1	2	4	1	1

Matrix representation of C₅⋊C₈⋊₇D₄ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	35	40	0	0
0	0	36	40	0	0
0	0	0	0	0	7
0	0	0	0	35	6

32	0	0	0	0	0
0	32	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	22	0	0
0	0	13	0	0	0

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

0	1	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	0	0	0	0	1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,36,0,0,0,0,40,40,0,0,0,0,0,0,0,35,0,0,0,0,7,6],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,13,0,0,0,0,22,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₅⋊C₈⋊₇D₄ in GAP, Magma, Sage, TeX

C_5\rtimes C_8\rtimes_7D_4

% in TeX

G:=Group("C5:C8:7D4");

// GroupNames label

G:=SmallGroup(320,1111);

// by ID

G=gap.SmallGroup(320,1111);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,6278,1595]);

// Polycyclic

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

// generators/relations

G = C5⋊C8⋊7D4 order 320 = 26·5

4th semidirect product of C5⋊C8 and D4 acting via D4/C22=C2

G = C₅⋊C₈⋊₇D₄ order 320 = 2⁶·5

4^th semidirect product of C₅⋊C₈ and D₄ acting via D₄/C₂²=C₂