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G = C40⋊D4order 320 = 26·5

17th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4017D4, D106M4(2), C812(C5⋊D4), C511(C89D4), C408C428C2, (C2×C8).188D10, C10.109(C4×D4), C20.442(C2×D4), (C2×M4(2))⋊7D5, D101C838C2, C23.19(C4×D5), C10.56(C8○D4), (C10×M4(2))⋊6C2, C20.8Q840C2, C2.21(D5×M4(2)), C23.D5.19C4, D10⋊C4.26C4, C20.253(C4○D4), C4.137(C4○D20), C20.55D429C2, (C2×C40).234C22, (C2×C20).867C23, C10.D4.26C4, (C22×C4).136D10, C10.66(C2×M4(2)), C2.17(D20.2C4), (C22×C20).374C22, (C4×Dic5).209C22, (D5×C2×C8)⋊27C2, (C2×C4).50(C4×D5), C2.24(C4×C5⋊D4), (C4×C5⋊D4).16C2, (C2×C5⋊D4).19C4, C4.133(C2×C5⋊D4), C22.146(C2×C4×D5), (C2×C20).360(C2×C4), (C2×C4×D5).355C22, (C2×Dic5).36(C2×C4), (C22×D5).81(C2×C4), (C2×C4).809(C22×D5), (C22×C10).135(C2×C4), (C2×C10).238(C22×C4), (C2×C52C8).331C22, SmallGroup(320,754)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C40⋊D4
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C40⋊D4
C5C2×C10 — C40⋊D4
C1C2×C4C2×M4(2)

Generators and relations for C40⋊D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a9, cbc=b-1 >

Subgroups: 382 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C40, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C89D4, C8×D5 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C5×M4(2) [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C408C4, D101C8, C20.55D4, D5×C2×C8, C4×C5⋊D4, C10×M4(2), C40⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C89D4, C2×C4×D5, C4○D20, C2×C5⋊D4, D5×M4(2), D20.2C4, C4×C5⋊D4, C40⋊D4

Smallest permutation representation of C40⋊D4
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)
(1 135 43 103)(2 124 44 92)(3 153 45 81)(4 142 46 110)(5 131 47 99)(6 160 48 88)(7 149 49 117)(8 138 50 106)(9 127 51 95)(10 156 52 84)(11 145 53 113)(12 134 54 102)(13 123 55 91)(14 152 56 120)(15 141 57 109)(16 130 58 98)(17 159 59 87)(18 148 60 116)(19 137 61 105)(20 126 62 94)(21 155 63 83)(22 144 64 112)(23 133 65 101)(24 122 66 90)(25 151 67 119)(26 140 68 108)(27 129 69 97)(28 158 70 86)(29 147 71 115)(30 136 72 104)(31 125 73 93)(32 154 74 82)(33 143 75 111)(34 132 76 100)(35 121 77 89)(36 150 78 118)(37 139 79 107)(38 128 80 96)(39 157 41 85)(40 146 42 114)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 65)(42 74)(44 52)(45 61)(46 70)(47 79)(49 57)(50 66)(51 75)(54 62)(55 71)(56 80)(59 67)(60 76)(64 72)(69 77)(81 137)(82 146)(83 155)(84 124)(85 133)(86 142)(87 151)(88 160)(89 129)(90 138)(91 147)(92 156)(93 125)(94 134)(95 143)(96 152)(97 121)(98 130)(99 139)(100 148)(101 157)(102 126)(103 135)(104 144)(105 153)(106 122)(107 131)(108 140)(109 149)(110 158)(111 127)(112 136)(113 145)(114 154)(115 123)(116 132)(117 141)(118 150)(119 159)(120 128)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444445588888888888810···101010101020···202020202040···40
size1111410101111410102020222222441010101020202···244442···244444···4

68 irreducible representations

dim1111111111112222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4M4(2)D10D10C8○D4C5⋊D4C4×D5C4×D5C4○D20D5×M4(2)D20.2C4
kernelC40⋊D4C20.8Q8C408C4D101C8C20.55D4D5×C2×C8C4×C5⋊D4C10×M4(2)C10.D4D10⋊C4C23.D5C2×C5⋊D4C40C2×M4(2)C20D10C2×C8C22×C4C10C8C2×C4C23C4C2C2
# reps1111111122222224424844844

Matrix representation of C40⋊D4 in GL4(𝔽41) generated by

03400
6600
00118
003530
,
20300
32100
001121
00630
,
6700
363500
0010
0001
G:=sub<GL(4,GF(41))| [0,6,0,0,34,6,0,0,0,0,11,35,0,0,8,30],[20,3,0,0,3,21,0,0,0,0,11,6,0,0,21,30],[6,36,0,0,7,35,0,0,0,0,1,0,0,0,0,1] >;

C40⋊D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes D_4
% in TeX

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

G:=SmallGroup(320,754);
// by ID

G=gap.SmallGroup(320,754);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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