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G = D409C4order 320 = 26·5

3rd semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D409C4, C42.17D10, C82(C4×D5), C4012(C2×C4), (C4×D20)⋊2C2, C8⋊C42D5, C406C42C2, D2019(C2×C4), C52(D8⋊C4), (C2×D40).7C2, C10.42(C4×D4), C2.15(C4×D20), (C2×C8).54D10, (C2×C4).115D20, (C2×C20).237D4, D205C437C2, C2.2(C8⋊D10), C10.3(C8⋊C22), (C2×C40).55C22, (C4×C20).15C22, C22.31(C2×D20), C20.223(C4○D4), C4.107(C4○D20), C20.165(C22×C4), (C2×C20).732C23, (C2×D20).196C22, C4⋊Dic5.266C22, C4.64(C2×C4×D5), (C5×C8⋊C4)⋊3C2, (C2×C10).115(C2×D4), (C2×C4).676(C22×D5), SmallGroup(320,338)

Series: Derived Chief Lower central Upper central

C1C20 — D409C4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — D409C4
C5C10C20 — D409C4
C1C22C42C8⋊C4

Generators and relations for D409C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a20b >

Subgroups: 662 in 132 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×8], C2×C10, C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C40 [×2], C40, C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], D8⋊C4, D40 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20 [×2], C406C4, D205C4 [×2], C5×C8⋊C4, C4×D20 [×2], C2×D40, D409C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22 [×2], C4×D5 [×2], D20 [×2], C22×D5, D8⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8⋊D10 [×2], D409C4

Smallest permutation representation of D409C4
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)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 130)(10 129)(11 128)(12 127)(13 126)(14 125)(15 124)(16 123)(17 122)(18 121)(19 160)(20 159)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 140)(40 139)(41 84)(42 83)(43 82)(44 81)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)
(1 119 124 52)(2 100 125 73)(3 81 126 54)(4 102 127 75)(5 83 128 56)(6 104 129 77)(7 85 130 58)(8 106 131 79)(9 87 132 60)(10 108 133 41)(11 89 134 62)(12 110 135 43)(13 91 136 64)(14 112 137 45)(15 93 138 66)(16 114 139 47)(17 95 140 68)(18 116 141 49)(19 97 142 70)(20 118 143 51)(21 99 144 72)(22 120 145 53)(23 101 146 74)(24 82 147 55)(25 103 148 76)(26 84 149 57)(27 105 150 78)(28 86 151 59)(29 107 152 80)(30 88 153 61)(31 109 154 42)(32 90 155 63)(33 111 156 44)(34 92 157 65)(35 113 158 46)(36 94 159 67)(37 115 160 48)(38 96 121 69)(39 117 122 50)(40 98 123 71)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order122222224···4444455888810···1020···2020···2040···40
size1111202020202···2202020202244442···22···24···44···4

62 irreducible representations

dim11111112222222244
type+++++++++++++
imageC1C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5D20C4○D20C8⋊C22C8⋊D10
kernelD409C4C406C4D205C4C5×C8⋊C4C4×D20C2×D40D40C2×C20C8⋊C4C20C42C2×C8C8C2×C4C4C10C2
# reps11212182222488828

Matrix representation of D409C4 in GL6(𝔽41)

3910000
3620000
00717151
001593814
0021424
003213923
,
100000
4400000
00271400
00301400
006151630
001151225
,
900000
090000
00102033
0027380
00252362
0038351529

G:=sub<GL(6,GF(41))| [39,36,0,0,0,0,1,2,0,0,0,0,0,0,7,15,2,3,0,0,17,9,14,21,0,0,15,38,2,39,0,0,1,14,4,23],[1,4,0,0,0,0,0,40,0,0,0,0,0,0,27,30,6,1,0,0,14,14,15,15,0,0,0,0,16,12,0,0,0,0,30,25],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,10,2,25,38,0,0,20,7,2,35,0,0,3,38,36,15,0,0,3,0,2,29] >;

D409C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_9C_4
% in TeX

G:=Group("D40:9C4");
// GroupNames label

G:=SmallGroup(320,338);
// by ID

G=gap.SmallGroup(320,338);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,387,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^20*b>;
// generators/relations

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