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## G = C5×C8⋊6D4order 320 = 26·5

### Direct product of C5 and C8⋊6D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C8⋊6D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C40 — C5×C22⋊C8 — C5×C8⋊6D4
 Lower central C1 — C22 — C5×C8⋊6D4
 Upper central C1 — C2×C20 — C5×C8⋊6D4

Generators and relations for C5×C86D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C40 [×2], C40 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C86D4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×C20 [×2], D4×C10, C4×C40, C5×C22⋊C8 [×2], C5×C4⋊C8, D4×C20, C10×M4(2) [×2], C5×C86D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], M4(2) [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×M4(2), C8○D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C86D4, C5×M4(2) [×2], C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×M4(2), C5×C8○D4, C5×C86D4

Smallest permutation representation of C5×C86D4
On 160 points
Generators in S160
(1 51 14 57 17)(2 52 15 58 18)(3 53 16 59 19)(4 54 9 60 20)(5 55 10 61 21)(6 56 11 62 22)(7 49 12 63 23)(8 50 13 64 24)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 157 116 149 108)(42 158 117 150 109)(43 159 118 151 110)(44 160 119 152 111)(45 153 120 145 112)(46 154 113 146 105)(47 155 114 147 106)(48 156 115 148 107)(89 122 142 101 129)(90 123 143 102 130)(91 124 144 103 131)(92 125 137 104 132)(93 126 138 97 133)(94 127 139 98 134)(95 128 140 99 135)(96 121 141 100 136)
(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)
(1 105 27 101)(2 106 28 102)(3 107 29 103)(4 108 30 104)(5 109 31 97)(6 110 32 98)(7 111 25 99)(8 112 26 100)(9 157 78 92)(10 158 79 93)(11 159 80 94)(12 160 73 95)(13 153 74 96)(14 154 75 89)(15 155 76 90)(16 156 77 91)(17 146 67 142)(18 147 68 143)(19 148 69 144)(20 149 70 137)(21 150 71 138)(22 151 72 139)(23 152 65 140)(24 145 66 141)(33 128 63 119)(34 121 64 120)(35 122 57 113)(36 123 58 114)(37 124 59 115)(38 125 60 116)(39 126 61 117)(40 127 62 118)(41 85 132 54)(42 86 133 55)(43 87 134 56)(44 88 135 49)(45 81 136 50)(46 82 129 51)(47 83 130 52)(48 84 131 53)
(1 101)(2 98)(3 103)(4 100)(5 97)(6 102)(7 99)(8 104)(9 96)(10 93)(11 90)(12 95)(13 92)(14 89)(15 94)(16 91)(17 142)(18 139)(19 144)(20 141)(21 138)(22 143)(23 140)(24 137)(25 111)(26 108)(27 105)(28 110)(29 107)(30 112)(31 109)(32 106)(33 119)(34 116)(35 113)(36 118)(37 115)(38 120)(39 117)(40 114)(41 81)(42 86)(43 83)(44 88)(45 85)(46 82)(47 87)(48 84)(49 135)(50 132)(51 129)(52 134)(53 131)(54 136)(55 133)(56 130)(57 122)(58 127)(59 124)(60 121)(61 126)(62 123)(63 128)(64 125)(65 152)(66 149)(67 146)(68 151)(69 148)(70 145)(71 150)(72 147)(73 160)(74 157)(75 154)(76 159)(77 156)(78 153)(79 158)(80 155)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 5C 5D 8A ··· 8H 8I 8J 8K 8L 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AF 20AG ··· 20AN 40A ··· 40AF 40AG ··· 40AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C10 C20 C20 C20 D4 M4(2) C4○D4 C8○D4 C5×D4 C5×M4(2) C5×C4○D4 C5×C8○D4 kernel C5×C8⋊6D4 C4×C40 C5×C22⋊C8 C5×C4⋊C8 D4×C20 C10×M4(2) C5×C22⋊C4 C5×C4⋊C4 D4×C10 C8⋊6D4 C4×C8 C22⋊C8 C4⋊C8 C4×D4 C2×M4(2) C22⋊C4 C4⋊C4 C2×D4 C40 C20 C20 C10 C8 C4 C4 C2 # reps 1 1 2 1 1 2 4 2 2 4 4 8 4 4 8 16 8 8 2 4 2 4 8 16 8 16

Matrix representation of C5×C86D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 16 39 0 0 21 25 0 0 0 0 0 1 0 0 9 0
,
 34 6 0 0 19 7 0 0 0 0 1 0 0 0 0 1
,
 34 6 0 0 33 7 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,21,0,0,39,25,0,0,0,0,0,9,0,0,1,0],[34,19,0,0,6,7,0,0,0,0,1,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;

C5×C86D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_6D_4
% in TeX

G:=Group("C5xC8:6D4");
// GroupNames label

G:=SmallGroup(320,937);
// by ID

G=gap.SmallGroup(320,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1731,436,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=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^5,d*c*d=c^-1>;
// generators/relations

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