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G = D4017C4order 320 = 26·5

The semidirect product of D40 and C4 acting through Inn(D40)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4017C4, C40.67D4, C8.14D20, Dic2017C4, C42.267D10, (C4×C8)⋊10D5, C54(C8○D8), (C4×C40)⋊15C2, C8.22(C4×D5), C40⋊C211C4, C40.93(C2×C4), C10.39(C4×D4), C2.12(C4×D20), C4.76(C2×D20), D20.27(C2×C4), (C2×C8).324D10, C20.296(C2×D4), C40.6C417C2, D204C415C2, D407C2.10C2, D20.3C411C2, (C2×C20).791C23, C20.163(C22×C4), (C2×C40).407C22, (C4×C20).328C22, Dic10.28(C2×C4), C4○D20.35C22, C22.20(C4○D20), C4.Dic5.33C22, C4.62(C2×C4×D5), (C2×C10).62(C4○D4), (C2×C4).672(C22×D5), SmallGroup(320,327)

Series: Derived Chief Lower central Upper central

C1C20 — D4017C4
C1C5C10C20C2×C20C4○D20D407C2 — D4017C4
C5C10C20 — D4017C4
C1C8C2×C8C4×C8

Generators and relations for D4017C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, ac=ca, cbc-1=a10b >

Subgroups: 374 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], C2×C10, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8 [×2], C40 [×4], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5 [×2], C4×C20, C2×C40 [×2], C4○D20 [×2], D204C4 [×2], C40.6C4, C4×C40, D20.3C4 [×2], D407C2, D4017C4
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, C4×D5 [×2], D20 [×2], C22×D5, C8○D8, C2×C4×D5, C2×D20, C4○D20, C4×D20, D4017C4

Smallest permutation representation of D4017C4
On 80 points
Generators in S80
(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)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)
(1 31 21 11)(2 32 22 12)(3 33 23 13)(4 34 24 14)(5 35 25 15)(6 36 26 16)(7 37 27 17)(8 38 28 18)(9 39 29 19)(10 40 30 20)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)

G:=sub<Sym(80)| (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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,31,21,11)(2,32,22,12)(3,33,23,13)(4,34,24,14)(5,35,25,15)(6,36,26,16)(7,37,27,17)(8,38,28,18)(9,39,29,19)(10,40,30,20)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)>;

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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,31,21,11)(2,32,22,12)(3,33,23,13)(4,34,24,14)(5,35,25,15)(6,36,26,16)(7,37,27,17)(8,38,28,18)(9,39,29,19)(10,40,30,20)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80) );

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)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52)], [(1,31,21,11),(2,32,22,12),(3,33,23,13),(4,34,24,14),(5,35,25,15),(6,36,26,16),(7,37,27,17),(8,38,28,18),(9,39,29,19),(10,40,30,20),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)])

92 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I5A5B8A8B8C8D8E···8J8K8L8M8N10A···10F20A···20X40A···40AF
order12222444···4445588888···8888810···1020···2040···40
size1122020112···220202211112···2202020202···22···22···2

92 irreducible representations

dim1111111112222222222
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10D10C4×D5D20C8○D8C4○D20D4017C4
kernelD4017C4D204C4C40.6C4C4×C40D20.3C4D407C2C40⋊C2D40Dic20C40C4×C8C2×C10C42C2×C8C8C8C5C22C1
# reps12112142222224888832

Matrix representation of D4017C4 in GL2(𝔽41) generated by

280
022
,
022
280
,
320
040
G:=sub<GL(2,GF(41))| [28,0,0,22],[0,28,22,0],[32,0,0,40] >;

D4017C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{17}C_4
% in TeX

G:=Group("D40:17C4");
// GroupNames label

G:=SmallGroup(320,327);
// by ID

G=gap.SmallGroup(320,327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^10*b>;
// generators/relations

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