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G = D5:C4wrC2order 320 = 26·5

The semidirect product of D5 and C4wrC2 acting via C4wrC2/C4oD4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5:C4wrC2, (D4xD5):8C4, C4oD4:1F5, D4:5(C2xF5), Q8:5(C2xF5), (Q8xD5):8C4, C4oD20:1C4, D20:5(C2xC4), D4:F5:7C2, Q8:2F5:7C2, (C4xF5):7C22, C4.F5:5C22, Dic10:5(C2xC4), D10.10(C2xD4), (C4xD5).121D4, D5:M4(2):5C2, C4.21(C22xF5), C20.21(C22xC4), (C4xD5).43C23, (C22xD5).71D4, C4.46(C22:F5), C20.46(C22:C4), (C2xDic5).123D4, Dic5.114(C2xD4), C22.4(C22:F5), D10.48(C22:C4), D4:2D5.13C22, Q8:2D5.13C22, Dic5.15(C22:C4), C5:3(C2xC4wrC2), (C2xC4xF5):4C2, (C5xC4oD4):1C4, (C5xD4):5(C2xC4), (C5xQ8):5(C2xC4), (D5xC4oD4).5C2, (C2xC4).89(C2xF5), (C2xC20).66(C2xC4), (C4xD5).27(C2xC4), C2.34(C2xC22:F5), C10.33(C2xC22:C4), (C2xC4xD5).210C22, (C2xC10).4(C22:C4), SmallGroup(320,1130)

Series: Derived Chief Lower central Upper central

C1C20 — D5:C4wrC2
C1C5C10D10C4xD5C4xF5C2xC4xF5 — D5:C4wrC2
C5C10C20 — D5:C4wrC2
C1C4C2xC4C4oD4

Generators and relations for D5:C4wrC2
 G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=a2b, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 730 in 170 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, F5, D10, D10, C2xC10, C2xC10, C4wrC2, C2xC42, C2xM4(2), C2xC4oD4, C5:C8, Dic10, Dic10, C4xD5, C4xD5, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C2xF5, C22xD5, C22xD5, C2xC4wrC2, D5:C8, C4.F5, C4xF5, C4xF5, C22.F5, C2xC4xD5, C2xC4xD5, C4oD20, C4oD20, D4xD5, D4xD5, D4:2D5, D4:2D5, Q8xD5, Q8:2D5, C5xC4oD4, C22xF5, D4:F5, Q8:2F5, D5:M4(2), C2xC4xF5, D5xC4oD4, D5:C4wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, F5, C4wrC2, C2xC22:C4, C2xF5, C2xC4wrC2, C22:F5, C22xF5, C2xC22:F5, D5:C4wrC2

Smallest permutation representation of D5:C4wrC2
On 40 points
Generators in S40
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 9)(2 6 5 7)(3 8 4 10)(11 18 12 20)(13 17 15 16)(14 19)(21 38 27 35)(22 40 26 33)(23 37 30 31)(24 39 29 34)(25 36 28 32)

G:=sub<Sym(40)| (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,20)(13,17,15,16)(14,19)(21,38,27,35)(22,40,26,33)(23,37,30,31)(24,39,29,34)(25,36,28,32)>;

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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,20)(13,17,15,16)(14,19)(21,38,27,35)(22,40,26,33)(23,37,30,31)(24,39,29,34)(25,36,28,32) );

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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,26),(12,27),(13,28),(14,29),(15,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,9),(2,6,5,7),(3,8,4,10),(11,18,12,20),(13,17,15,16),(14,19),(21,38,27,35),(22,40,26,33),(23,37,30,31),(24,39,29,34),(25,36,28,32)]])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4O4P 5 8A8B8C8D10A10B10C10D20A20B20C20D20E
order122222224444444···4458888101010102020202020
size112455102011245510···1020420202020488844888

38 irreducible representations

dim111111111122224444448
type+++++++++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4D4C4wrC2F5C2xF5C2xF5C2xF5C22:F5C22:F5D5:C4wrC2
kernelD5:C4wrC2D4:F5Q8:2F5D5:M4(2)C2xC4xF5D5xC4oD4C4oD20D4xD5Q8xD5C5xC4oD4C4xD5C2xDic5C22xD5D5C4oD4C2xC4D4Q8C4C22C1
# reps122111222221181111222

Matrix representation of D5:C4wrC2 in GL6(F41)

100000
010000
000100
000010
000001
0040404040
,
4000000
0400000
000100
001000
0040404040
000001
,
32320000
090000
001000
000100
000010
000001
,
990000
23320000
001000
000100
000010
000001
,
40360000
090000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,23,0,0,0,0,9,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,36,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D5:C4wrC2 in GAP, Magma, Sage, TeX

D_5\rtimes C_4\wr C_2
% in TeX

G:=Group("D5:C4wrC2");
// GroupNames label

G:=SmallGroup(320,1130);
// by ID

G=gap.SmallGroup(320,1130);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^3,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
// generators/relations

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