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## G = C8⋊(C4×D5)  order 320 = 26·5

### 3rd semidirect product of C8 and C4×D5 acting via C4×D5/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C8⋊(C4×D5)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×C8⋊D5 — C8⋊(C4×D5)
 Lower central C5 — C10 — C20 — C8⋊(C4×D5)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8⋊(C4×D5)
G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a3, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×4], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, M4(2)⋊C4, C8⋊D5 [×4], C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C405C4, C5×C4.Q8, D5×C4⋊C4, C4⋊C47D5, C2×C8⋊D5, C8⋊(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D40⋊C2, SD16⋊D5, C8⋊(C4×D5)

Smallest permutation representation of C8⋊(C4×D5)
On 160 points
Generators in S160
(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 55 28 155)(2 50 29 158)(3 53 30 153)(4 56 31 156)(5 51 32 159)(6 54 25 154)(7 49 26 157)(8 52 27 160)(9 63 102 23)(10 58 103 18)(11 61 104 21)(12 64 97 24)(13 59 98 19)(14 62 99 22)(15 57 100 17)(16 60 101 20)(33 86 150 139)(34 81 151 142)(35 84 152 137)(36 87 145 140)(37 82 146 143)(38 85 147 138)(39 88 148 141)(40 83 149 144)(41 109 79 131)(42 112 80 134)(43 107 73 129)(44 110 74 132)(45 105 75 135)(46 108 76 130)(47 111 77 133)(48 106 78 136)(65 124 117 95)(66 127 118 90)(67 122 119 93)(68 125 120 96)(69 128 113 91)(70 123 114 94)(71 126 115 89)(72 121 116 92)
(1 39 133 23 113)(2 40 134 24 114)(3 33 135 17 115)(4 34 136 18 116)(5 35 129 19 117)(6 36 130 20 118)(7 37 131 21 119)(8 38 132 22 120)(9 91 55 88 47)(10 92 56 81 48)(11 93 49 82 41)(12 94 50 83 42)(13 95 51 84 43)(14 96 52 85 44)(15 89 53 86 45)(16 90 54 87 46)(25 145 108 60 66)(26 146 109 61 67)(27 147 110 62 68)(28 148 111 63 69)(29 149 112 64 70)(30 150 105 57 71)(31 151 106 58 72)(32 152 107 59 65)(73 98 124 159 137)(74 99 125 160 138)(75 100 126 153 139)(76 101 127 154 140)(77 102 128 155 141)(78 103 121 156 142)(79 104 122 157 143)(80 97 123 158 144)
(1 113)(2 118)(3 115)(4 120)(5 117)(6 114)(7 119)(8 116)(9 88)(10 85)(11 82)(12 87)(13 84)(14 81)(15 86)(16 83)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 70)(26 67)(27 72)(28 69)(29 66)(30 71)(31 68)(32 65)(42 46)(44 48)(49 93)(50 90)(51 95)(52 92)(53 89)(54 94)(55 91)(56 96)(57 150)(58 147)(59 152)(60 149)(61 146)(62 151)(63 148)(64 145)(74 78)(76 80)(97 140)(98 137)(99 142)(100 139)(101 144)(102 141)(103 138)(104 143)(106 110)(108 112)(121 160)(122 157)(123 154)(124 159)(125 156)(126 153)(127 158)(128 155)(130 134)(132 136)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 2 2 4 4 4 4 10 10 20 20 20 20 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 D10 D10 C4×D5 C8⋊C22 C8.C22 Q8×D5 D4×D5 D40⋊C2 SD16⋊D5 kernel C8⋊(C4×D5) C10.D8 C20.Q8 C40⋊5C4 C5×C4.Q8 D5×C4⋊C4 C4⋊C4⋊7D5 C2×C8⋊D5 C8⋊D5 C4×D5 C2×Dic5 C22×D5 C4.Q8 C4⋊C4 C2×C8 C8 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 2 4 2 8 1 1 2 2 4 4

Matrix representation of C8⋊(C4×D5) in GL6(𝔽41)

 35 2 0 0 0 0 2 6 0 0 0 0 0 0 32 15 9 26 0 0 26 9 15 32 0 0 32 15 32 15 0 0 26 9 26 9
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 27 0 7 0 0 0 0 27 0 7 0 0 7 0 14 0 0 0 0 7 0 14
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 40 0 0 0 0 1 0 0 0 0 0 0 0 34 40 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 34 40 0 0 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 7 7

G:=sub<GL(6,GF(41))| [35,2,0,0,0,0,2,6,0,0,0,0,0,0,32,26,32,26,0,0,15,9,15,9,0,0,9,15,32,26,0,0,26,32,15,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,27,0,7,0,0,0,0,27,0,7,0,0,7,0,14,0,0,0,0,7,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,40,7] >;

C8⋊(C4×D5) in GAP, Magma, Sage, TeX

C_8\rtimes (C_4\times D_5)
% in TeX

G:=Group("C8:(C4xD5)");
// GroupNames label

G:=SmallGroup(320,488);
// by ID

G=gap.SmallGroup(320,488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,555,58,438,102,12550]);
// Polycyclic

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

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