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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 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10 [×2], 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 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], 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], C2×C40 [×2], 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 [×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, C8⋊D5 [×2], 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 81 149 64)(2 110 150 53)(3 99 151 42)(4 88 152 71)(5 117 153 60)(6 106 154 49)(7 95 155 78)(8 84 156 67)(9 113 157 56)(10 102 158 45)(11 91 159 74)(12 120 160 63)(13 109 121 52)(14 98 122 41)(15 87 123 70)(16 116 124 59)(17 105 125 48)(18 94 126 77)(19 83 127 66)(20 112 128 55)(21 101 129 44)(22 90 130 73)(23 119 131 62)(24 108 132 51)(25 97 133 80)(26 86 134 69)(27 115 135 58)(28 104 136 47)(29 93 137 76)(30 82 138 65)(31 111 139 54)(32 100 140 43)(33 89 141 72)(34 118 142 61)(35 107 143 50)(36 96 144 79)(37 85 145 68)(38 114 146 57)(39 103 147 46)(40 92 148 75)
(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 94)(42 83)(43 112)(44 101)(45 90)(46 119)(47 108)(48 97)(49 86)(50 115)(51 104)(52 93)(53 82)(54 111)(55 100)(56 89)(57 118)(58 107)(59 96)(60 85)(61 114)(62 103)(63 92)(64 81)(65 110)(66 99)(67 88)(68 117)(69 106)(70 95)(71 84)(72 113)(73 102)(74 91)(75 120)(76 109)(77 98)(78 87)(79 116)(80 105)(121 137)(122 126)(123 155)(124 144)(125 133)(127 151)(128 140)(130 158)(131 147)(132 136)(134 154)(135 143)(138 150)(141 157)(142 146)(145 153)(148 160)(152 156)

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

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

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

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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