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G = D4012C4order 320 = 26·5

6th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4012C4, Dic55D8, C54(C4×D8), C87(C4×D5), C2.3(D5×D8), C4014(C2×C4), D2017(C2×C4), C2.D813D5, (C8×Dic5)⋊3C2, (C2×D40).9C2, C10.82(C4×D4), C10.26(C2×D8), D208C47C2, C4⋊C4.166D10, (C2×C8).225D10, D206C419C2, C22.87(D4×D5), C20.35(C4○D4), C10.73(C4○D8), (C2×C40).77C22, C4.7(Q82D5), C2.3(Q8.D10), C20.105(C22×C4), (C2×C20).288C23, (C2×Dic5).275D4, (C2×D20).84C22, C2.12(D208C4), (C4×Dic5).262C22, C4.43(C2×C4×D5), (C5×C2.D8)⋊2C2, (C2×C10).293(C2×D4), (C5×C4⋊C4).81C22, (C2×C4).391(C22×D5), (C2×C52C8).237C22, SmallGroup(320,499)

Series: Derived Chief Lower central Upper central

C1C20 — D4012C4
C1C5C10C20C2×C20C4×Dic5D208C4 — D4012C4
C5C10C20 — D4012C4
C1C22C2×C4C2.D8

Generators and relations for D4012C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a30b >

Subgroups: 646 in 134 conjugacy classes, 51 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×6], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×8], C2×C10, C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C52C8, C40 [×2], C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×D8, D40 [×4], C2×C52C8, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5 [×2], C2×D20 [×2], D206C4 [×2], C8×Dic5, C5×C2.D8, D208C4 [×2], C2×D40, D4012C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], C22×D5, C4×D8, C2×C4×D5, D4×D5, Q82D5, D208C4, D5×D8, Q8.D10, D4012C4

Smallest permutation representation of D4012C4
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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 83)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
(1 95 62 123)(2 86 63 154)(3 117 64 145)(4 108 65 136)(5 99 66 127)(6 90 67 158)(7 81 68 149)(8 112 69 140)(9 103 70 131)(10 94 71 122)(11 85 72 153)(12 116 73 144)(13 107 74 135)(14 98 75 126)(15 89 76 157)(16 120 77 148)(17 111 78 139)(18 102 79 130)(19 93 80 121)(20 84 41 152)(21 115 42 143)(22 106 43 134)(23 97 44 125)(24 88 45 156)(25 119 46 147)(26 110 47 138)(27 101 48 129)(28 92 49 160)(29 83 50 151)(30 114 51 142)(31 105 52 133)(32 96 53 124)(33 87 54 155)(34 118 55 146)(35 109 56 137)(36 100 57 128)(37 91 58 159)(38 82 59 150)(39 113 60 141)(40 104 61 132)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444444444558888888810···102020202020···2040···40
size11112020202022444455551010222222101010102···244448···84···4

56 irreducible representations

dim1111111222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C4D4D5D8C4○D4D10D10C4○D8C4×D5Q82D5D4×D5D5×D8Q8.D10
kernelD4012C4D206C4C8×Dic5C5×C2.D8D208C4C2×D40D40C2×Dic5C2.D8Dic5C20C4⋊C4C2×C8C10C8C4C22C2C2
# reps1211218224242482244

Matrix representation of D4012C4 in GL4(𝔽41) generated by

343400
7100
00017
001217
,
403400
0100
0010
00140
,
9000
0900
003011
001511
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,0,12,0,0,17,17],[40,0,0,0,34,1,0,0,0,0,1,1,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11] >;

D4012C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{12}C_4
% in TeX

G:=Group("D40:12C4");
// GroupNames label

G:=SmallGroup(320,499);
// by ID

G=gap.SmallGroup(320,499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,100,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^30*b>;
// generators/relations

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