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## G = C5×C8⋊4D4order 320 = 26·5

### Direct product of C5 and C8⋊4D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C10×D8 — C5×C8⋊4D4
 Lower central C1 — C2 — C2×C4 — C5×C8⋊4D4
 Upper central C1 — C2×C10 — C4×C20 — C5×C8⋊4D4

Generators and relations for C5×C84D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 386 in 162 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C10, C42, C2×C8, D8, C2×D4, C2×D4, C20, C2×C10, C2×C10, C4×C8, C41D4, C2×D8, C40, C2×C20, C2×C20, C5×D4, C22×C10, C84D4, C4×C20, C2×C40, C5×D8, D4×C10, D4×C10, C4×C40, C5×C41D4, C10×D8, C5×C84D4
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C41D4, C2×D8, C5×D4, C22×C10, C84D4, C5×D8, D4×C10, C5×C41D4, C10×D8, C5×C84D4

Smallest permutation representation of C5×C84D4
On 160 points
Generators in S160
(1 139 57 131 49)(2 140 58 132 50)(3 141 59 133 51)(4 142 60 134 52)(5 143 61 135 53)(6 144 62 136 54)(7 137 63 129 55)(8 138 64 130 56)(9 25 99 17 91)(10 26 100 18 92)(11 27 101 19 93)(12 28 102 20 94)(13 29 103 21 95)(14 30 104 22 96)(15 31 97 23 89)(16 32 98 24 90)(33 107 44 127 117)(34 108 45 128 118)(35 109 46 121 119)(36 110 47 122 120)(37 111 48 123 113)(38 112 41 124 114)(39 105 42 125 115)(40 106 43 126 116)(65 82 155 73 147)(66 83 156 74 148)(67 84 157 75 149)(68 85 158 76 150)(69 86 159 77 151)(70 87 160 78 152)(71 88 153 79 145)(72 81 154 80 146)
(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)
(1 111 13 65)(2 112 14 66)(3 105 15 67)(4 106 16 68)(5 107 9 69)(6 108 10 70)(7 109 11 71)(8 110 12 72)(17 77 135 117)(18 78 136 118)(19 79 129 119)(20 80 130 120)(21 73 131 113)(22 74 132 114)(23 75 133 115)(24 76 134 116)(25 86 143 44)(26 87 144 45)(27 88 137 46)(28 81 138 47)(29 82 139 48)(30 83 140 41)(31 84 141 42)(32 85 142 43)(33 91 151 53)(34 92 152 54)(35 93 145 55)(36 94 146 56)(37 95 147 49)(38 96 148 50)(39 89 149 51)(40 90 150 52)(57 123 103 155)(58 124 104 156)(59 125 97 157)(60 126 98 158)(61 127 99 159)(62 128 100 160)(63 121 101 153)(64 122 102 154)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 152)(40 151)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 56)(50 55)(51 54)(52 53)(57 64)(58 63)(59 62)(60 61)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 112)(72 111)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(121 156)(122 155)(123 154)(124 153)(125 160)(126 159)(127 158)(128 157)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10AB 20A ··· 20X 40A ··· 40AF order 1 2 2 2 2 2 2 2 4 ··· 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 8 8 8 8 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D4 D8 C5×D4 C5×D4 C5×D8 kernel C5×C8⋊4D4 C4×C40 C5×C4⋊1D4 C10×D8 C8⋊4D4 C4×C8 C4⋊1D4 C2×D8 C40 C2×C20 C20 C8 C2×C4 C4 # reps 1 1 2 4 4 4 8 16 4 2 8 16 8 32

Matrix representation of C5×C84D4 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 29 12 0 0 29 29 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 40 0 0 0 0 0 1 39 0 0 1 40
,
 29 12 0 0 12 12 0 0 0 0 1 0 0 0 1 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[29,29,0,0,12,29,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,1,1,0,0,39,40],[29,12,0,0,12,12,0,0,0,0,1,1,0,0,0,40] >;

C5×C84D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_4D_4
% in TeX

G:=Group("C5xC8:4D4");
// GroupNames label

G:=SmallGroup(320,994);
// by ID

G=gap.SmallGroup(320,994);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,436,7004,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

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