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G = C405D4order 320 = 26·5

5th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C405D4, Dic51D8, (C2×D8)⋊3D5, (C10×D8)⋊5C2, C52C814D4, C87(C5⋊D4), C53(C84D4), C2.27(D5×D8), C4.20(D4×D5), (C2×D40)⋊18C2, C20⋊D44C2, (C8×Dic5)⋊5C2, C20.45(C2×D4), C10.44(C2×D8), (C2×D4).60D10, (C2×C8).236D10, (C2×C40).88C22, C22.252(D4×D5), C10.26(C41D4), C2.17(C20⋊D4), (C2×C20).428C23, (C2×Dic5).155D4, (D4×C10).78C22, (C2×D20).118C22, (C4×Dic5).269C22, C4.4(C2×C5⋊D4), (C2×D4⋊D5)⋊17C2, (C2×C10).341(C2×D4), (C2×C4).518(C22×D5), (C2×C52C8).277C22, SmallGroup(320,778)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C405D4
C1C5C10C20C2×C20C4×Dic5C20⋊D4 — C405D4
C5C10C2×C20 — C405D4
C1C22C2×C4C2×D8

Generators and relations for C405D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >

Subgroups: 878 in 162 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C42, C2×C8, C2×C8, D8 [×8], C2×D4 [×2], C2×D4 [×6], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C4×C8, C41D4 [×2], C2×D8, C2×D8 [×3], C52C8 [×2], C40 [×2], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×2], C22×C10 [×2], C84D4, D40 [×2], C2×C52C8, C4×Dic5, D4⋊D5 [×4], C2×C40, C5×D8 [×2], C2×D20 [×2], C2×C5⋊D4 [×4], D4×C10 [×2], C8×Dic5, C2×D40, C2×D4⋊D5 [×2], C20⋊D4 [×2], C10×D8, C405D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×4], C2×D4 [×3], D10 [×3], C41D4, C2×D8 [×2], C5⋊D4 [×2], C22×D5, C84D4, D4×D5 [×2], C2×C5⋊D4, D5×D8 [×2], C20⋊D4, C405D4

Smallest permutation representation of C405D4
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)
(1 115 44 124)(2 84 45 133)(3 93 46 142)(4 102 47 151)(5 111 48 160)(6 120 49 129)(7 89 50 138)(8 98 51 147)(9 107 52 156)(10 116 53 125)(11 85 54 134)(12 94 55 143)(13 103 56 152)(14 112 57 121)(15 81 58 130)(16 90 59 139)(17 99 60 148)(18 108 61 157)(19 117 62 126)(20 86 63 135)(21 95 64 144)(22 104 65 153)(23 113 66 122)(24 82 67 131)(25 91 68 140)(26 100 69 149)(27 109 70 158)(28 118 71 127)(29 87 72 136)(30 96 73 145)(31 105 74 154)(32 114 75 123)(33 83 76 132)(34 92 77 141)(35 101 78 150)(36 110 79 159)(37 119 80 128)(38 88 41 137)(39 97 42 146)(40 106 43 155)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 47)(42 46)(43 45)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 160)(120 159)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F10G···10N20A20B20C20D40A···40H
order12222222444444558888888810···1010···102020202040···40
size11118840402210101010222222101010102···28···844444···4

50 irreducible representations

dim11111122222222444
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D8D10D10C5⋊D4D4×D5D4×D5D5×D8
kernelC405D4C8×Dic5C2×D40C2×D4⋊D5C20⋊D4C10×D8C52C8C40C2×Dic5C2×D8Dic5C2×C8C2×D4C8C4C22C2
# reps11122122228248228

Matrix representation of C405D4 in GL4(𝔽41) generated by

03500
7600
001715
00300
,
211500
172000
0010
0001
,
6600
13500
0010
001840
G:=sub<GL(4,GF(41))| [0,7,0,0,35,6,0,0,0,0,17,30,0,0,15,0],[21,17,0,0,15,20,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,6,35,0,0,0,0,1,18,0,0,0,40] >;

C405D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_5D_4
% in TeX

G:=Group("C40:5D4");
// GroupNames label

G:=SmallGroup(320,778);
// by ID

G=gap.SmallGroup(320,778);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,422,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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