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G = C4029D4order 320 = 26·5

1st semidirect product of C40 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4029D4, C221D40, C23.25D20, (C2×C10)⋊5D8, (C2×D40)⋊9C2, C54(C87D4), (C22×C8)⋊6D5, C207D42C2, C813(C5⋊D4), C405C414C2, C10.17(C2×D8), (C2×C4).68D20, C2.17(C2×D40), D205C43C2, (C22×C40)⋊10C2, C20.413(C2×D4), (C2×C8).308D10, (C2×C20).356D4, C10.18(C4○D8), C4.112(C4○D20), C20.228(C4○D4), C10.71(C4⋊D4), C2.19(C207D4), (C2×C20).769C23, (C2×C40).380C22, (C2×D20).21C22, (C22×C10).141D4, C22.132(C2×D20), (C22×C4).431D10, C4⋊Dic5.24C22, C2.18(D407C2), (C22×C20).519C22, C4.106(C2×C5⋊D4), (C2×C10).159(C2×D4), (C2×C4).717(C22×D5), SmallGroup(320,742)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4029D4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C4029D4
C5C10C2×C20 — C4029D4
C1C22C22×C4C22×C8

Generators and relations for C4029D4
 G = < a,b,c | a40=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 694 in 134 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], C22×C4, C2×D4 [×4], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, C40 [×2], C40, D20 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C87D4, D40 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40 [×2], C2×C40 [×2], C2×D20 [×2], C2×C5⋊D4 [×2], C22×C20, C405C4, D205C4 [×2], C2×D40, C207D4 [×2], C22×C40, C4029D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, D20 [×2], C5⋊D4 [×2], C22×D5, C87D4, D40 [×2], C2×D20, C4○D20, C2×C5⋊D4, C2×D40, D407C2, C207D4, C4029D4

Smallest permutation representation of C4029D4
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 75 87 153)(2 74 88 152)(3 73 89 151)(4 72 90 150)(5 71 91 149)(6 70 92 148)(7 69 93 147)(8 68 94 146)(9 67 95 145)(10 66 96 144)(11 65 97 143)(12 64 98 142)(13 63 99 141)(14 62 100 140)(15 61 101 139)(16 60 102 138)(17 59 103 137)(18 58 104 136)(19 57 105 135)(20 56 106 134)(21 55 107 133)(22 54 108 132)(23 53 109 131)(24 52 110 130)(25 51 111 129)(26 50 112 128)(27 49 113 127)(28 48 114 126)(29 47 115 125)(30 46 116 124)(31 45 117 123)(32 44 118 122)(33 43 119 121)(34 42 120 160)(35 41 81 159)(36 80 82 158)(37 79 83 157)(38 78 84 156)(39 77 85 155)(40 76 86 154)
(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)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)

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

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

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

86 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A···8H10A···10N20A···20P40A···40AF
order12222222444444558···810···1020···2040···40
size111122404022224040222···22···22···22···2

86 irreducible representations

dim111111222222222222222
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D8D10D10C4○D8C5⋊D4D20D20C4○D20D40D407C2
kernelC4029D4C405C4D205C4C2×D40C207D4C22×C40C40C2×C20C22×C10C22×C8C20C2×C10C2×C8C22×C4C10C8C2×C4C23C4C22C2
# reps11212121122442484481616

Matrix representation of C4029D4 in GL4(𝔽41) generated by

352600
122300
00323
001829
,
1000
94000
002121
001820
,
1000
94000
00635
004035
G:=sub<GL(4,GF(41))| [35,12,0,0,26,23,0,0,0,0,3,18,0,0,23,29],[1,9,0,0,0,40,0,0,0,0,21,18,0,0,21,20],[1,9,0,0,0,40,0,0,0,0,6,40,0,0,35,35] >;

C4029D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{29}D_4
% in TeX

G:=Group("C40:29D4");
// GroupNames label

G:=SmallGroup(320,742);
// by ID

G=gap.SmallGroup(320,742);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1684,102,12550]);
// Polycyclic

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

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