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G = C4032D4order 320 = 26·5

4th semidirect product of C40 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4032D4, C815(C5⋊D4), C510(C89D4), C408C423C2, (C22×C8)⋊11D5, D101C83C2, (C22×C40)⋊16C2, (C2×C8).294D10, C10.108(C4×D4), C20.437(C2×D4), C20.8Q83C2, C23.35(C4×D5), C10.43(C8○D4), C222(C8⋊D5), (C2×C10)⋊11M4(2), C23.D5.16C4, D10⋊C4.13C4, C20.252(C4○D4), C4.136(C4○D20), C20.55D426C2, (C2×C20).861C23, (C2×C40).355C22, C10.D4.13C4, (C22×C4).402D10, C10.47(C2×M4(2)), C2.19(D20.3C4), (C22×C20).561C22, (C4×Dic5).207C22, (C2×C4).94(C4×D5), C2.23(C4×C5⋊D4), (C2×C8⋊D5)⋊23C2, C2.15(C2×C8⋊D5), (C2×C5⋊D4).18C4, (C4×C5⋊D4).15C2, C4.127(C2×C5⋊D4), C22.142(C2×C4×D5), (C2×C20).383(C2×C4), (C2×C4×D5).235C22, (C2×Dic5).34(C2×C4), (C22×D5).29(C2×C4), (C2×C4).803(C22×D5), (C2×C10).232(C22×C4), (C22×C10).164(C2×C4), (C2×C52C8).209C22, SmallGroup(320,738)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4032D4
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C4032D4
C5C2×C10 — C4032D4
C1C2×C4C22×C8

Generators and relations for C4032D4
 G = < a,b,c | a40=b4=c2=1, bab-1=cac=a29, cbc=b-1 >

Subgroups: 382 in 124 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C89D4, C8⋊D5, C2×C52C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C408C4, D101C8, C20.55D4, C2×C8⋊D5, C4×C5⋊D4, C22×C40, C4032D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C5⋊D4, C22×D5, C89D4, C8⋊D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C2×C8⋊D5, D20.3C4, C4×C5⋊D4, C4032D4

Smallest permutation representation of C4032D4
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 67 152 106)(2 56 153 95)(3 45 154 84)(4 74 155 113)(5 63 156 102)(6 52 157 91)(7 41 158 120)(8 70 159 109)(9 59 160 98)(10 48 121 87)(11 77 122 116)(12 66 123 105)(13 55 124 94)(14 44 125 83)(15 73 126 112)(16 62 127 101)(17 51 128 90)(18 80 129 119)(19 69 130 108)(20 58 131 97)(21 47 132 86)(22 76 133 115)(23 65 134 104)(24 54 135 93)(25 43 136 82)(26 72 137 111)(27 61 138 100)(28 50 139 89)(29 79 140 118)(30 68 141 107)(31 57 142 96)(32 46 143 85)(33 75 144 114)(34 64 145 103)(35 53 146 92)(36 42 147 81)(37 71 148 110)(38 60 149 99)(39 49 150 88)(40 78 151 117)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 112)(42 101)(43 90)(44 119)(45 108)(46 97)(47 86)(48 115)(49 104)(50 93)(51 82)(52 111)(53 100)(54 89)(55 118)(56 107)(57 96)(58 85)(59 114)(60 103)(61 92)(62 81)(63 110)(64 99)(65 88)(66 117)(67 106)(68 95)(69 84)(70 113)(71 102)(72 91)(73 120)(74 109)(75 98)(76 87)(77 116)(78 105)(79 94)(80 83)(121 133)(123 151)(124 140)(125 129)(126 158)(127 147)(128 136)(130 154)(131 143)(134 150)(135 139)(137 157)(138 146)(141 153)(144 160)(145 149)(148 156)(155 159)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A···8H8I8J8K8L10A···10N20A···20P40A···40AF
order1222222444444444558···8888810···1020···2040···40
size11112220111122202020222···2202020202···22···22···2

92 irreducible representations

dim1111111111112222222222222
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4M4(2)D10D10C8○D4C5⋊D4C4×D5C4×D5C4○D20C8⋊D5D20.3C4
kernelC4032D4C20.8Q8C408C4D101C8C20.55D4C2×C8⋊D5C4×C5⋊D4C22×C40C10.D4D10⋊C4C23.D5C2×C5⋊D4C40C22×C8C20C2×C10C2×C8C22×C4C10C8C2×C4C23C4C22C2
# reps111111112222222442484481616

Matrix representation of C4032D4 in GL4(𝔽41) generated by

31900
32900
001932
003723
,
174000
32400
003540
00356
,
1000
344000
0061
00635
G:=sub<GL(4,GF(41))| [31,32,0,0,9,9,0,0,0,0,19,37,0,0,32,23],[17,3,0,0,40,24,0,0,0,0,35,35,0,0,40,6],[1,34,0,0,0,40,0,0,0,0,6,6,0,0,1,35] >;

C4032D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{32}D_4
% in TeX

G:=Group("C40:32D4");
// GroupNames label

G:=SmallGroup(320,738);
// by ID

G=gap.SmallGroup(320,738);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,58,136,12550]);
// Polycyclic

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

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