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G = D5⋊C4≀C2order 320 = 26·5

The semidirect product of D5 and C4≀C2 acting via C4≀C2/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5⋊C4≀C2, (D4×D5)⋊8C4, C4○D41F5, D45(C2×F5), Q85(C2×F5), (Q8×D5)⋊8C4, C4○D201C4, D205(C2×C4), D4⋊F57C2, Q82F57C2, (C4×F5)⋊7C22, C4.F55C22, Dic105(C2×C4), D10.10(C2×D4), (C4×D5).121D4, D5⋊M4(2)⋊5C2, C4.21(C22×F5), C20.21(C22×C4), (C4×D5).43C23, (C22×D5).71D4, C4.46(C22⋊F5), C20.46(C22⋊C4), (C2×Dic5).123D4, Dic5.114(C2×D4), C22.4(C22⋊F5), D10.48(C22⋊C4), D42D5.13C22, Q82D5.13C22, Dic5.15(C22⋊C4), C53(C2×C4≀C2), (C2×C4×F5)⋊4C2, (C5×C4○D4)⋊1C4, (C5×D4)⋊5(C2×C4), (C5×Q8)⋊5(C2×C4), (D5×C4○D4).5C2, (C2×C4).89(C2×F5), (C2×C20).66(C2×C4), (C4×D5).27(C2×C4), C2.34(C2×C22⋊F5), C10.33(C2×C22⋊C4), (C2×C4×D5).210C22, (C2×C10).4(C22⋊C4), SmallGroup(320,1130)

Series: Derived Chief Lower central Upper central

C1C20 — D5⋊C4≀C2
C1C5C10D10C4×D5C4×F5C2×C4×F5 — D5⋊C4≀C2
C5C10C20 — D5⋊C4≀C2
C1C4C2×C4C4○D4

Generators and relations for D5⋊C4≀C2
 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 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42 [×3], C2×C8, M4(2) [×3], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5 [×6], C22×D5, C22×D5, C2×C4≀C2, D5⋊C8, C4.F5 [×2], C4×F5 [×2], C4×F5, C22.F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C22×F5, D4⋊F5 [×2], Q82F5 [×2], D5⋊M4(2), C2×C4×F5, D5×C4○D4, D5⋊C4≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4≀C2 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4≀C2, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, D5⋊C4≀C2

Smallest permutation representation of D5⋊C4≀C2
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+++++++++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4D4C4≀C2F5C2×F5C2×F5C2×F5C22⋊F5C22⋊F5D5⋊C4≀C2
kernelD5⋊C4≀C2D4⋊F5Q82F5D5⋊M4(2)C2×C4×F5D5×C4○D4C4○D20D4×D5Q8×D5C5×C4○D4C4×D5C2×Dic5C22×D5D5C4○D4C2×C4D4Q8C4C22C1
# reps122111222221181111222

Matrix representation of D5⋊C4≀C2 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⋊C4≀C2 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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