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G = C5⋊C2≀C4order 320 = 26·5

The semidirect product of C5 and C2≀C4 acting via C2≀C4/C22⋊C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C51C2≀C4, C23⋊F51C2, C22⋊C41F5, (C23×D5)⋊5C4, C23.F51C2, C23.1(C2×F5), (C2×Dic5).9D4, (C22×D5).9D4, C22⋊D20.1C2, C10.5(C23⋊C4), C2.8(D10.D4), C22.14(C22⋊F5), (C5×C22⋊C4)⋊1C4, (C22×C10).8(C2×C4), (C2×C5⋊D4).83C22, (C2×C10).14(C22⋊C4), SmallGroup(320,202)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C5⋊C2≀C4
C1C5C10C2×C10C22×D5C2×C5⋊D4C23⋊F5 — C5⋊C2≀C4
C5C10C2×C10C22×C10 — C5⋊C2≀C4
C1C2C22C23C22⋊C4

Generators and relations for C5⋊C2≀C4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a3, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 682 in 94 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4 [×3], D4 [×3], C23, C23 [×5], D5 [×3], C10, C10 [×2], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4 [×2], C24, Dic5, C20, F5, D10 [×10], C2×C10, C2×C10 [×2], C23⋊C4, C4.D4, C22≀C2, C5⋊C8, D20 [×2], C2×Dic5, C5⋊D4, C2×C20, C2×F5, C22×D5, C22×D5 [×4], C22×C10, C2≀C4, D10⋊C4, C5×C22⋊C4, C22.F5, C22⋊F5, C2×D20, C2×C5⋊D4, C23×D5, C23⋊F5, C23.F5, C22⋊D20, C5⋊C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C2≀C4, C22⋊F5, D10.D4, C5⋊C2≀C4

Character table of C5⋊C2≀C4

 class 12A2B2C2D2E2F4A4B4C4D58A8B10A10B10C10D10E20A20B20C20D
 size 1124202020820404044040444888888
ρ111111111111111111111111    trivial
ρ2111111111-1-11-1-1111111111    linear of order 2
ρ31111-11-1-11111-1-111111-1-1-1-1    linear of order 2
ρ41111-11-1-11-1-111111111-1-1-1-1    linear of order 2
ρ511111-11-1-1-ii1i-i11111-1-1-1-1    linear of order 4
ρ611111-11-1-1i-i1-ii11111-1-1-1-1    linear of order 4
ρ71111-1-1-11-1-ii1-ii111111111    linear of order 4
ρ81111-1-1-11-1i-i1i-i111111111    linear of order 4
ρ9222-20200-200200222-2-20000    orthogonal lifted from D4
ρ10222-20-200200200222-2-20000    orthogonal lifted from D4
ρ1144-400000000400-4-44000000    orthogonal lifted from C23⋊C4
ρ124-40020-2000040000-4000000    orthogonal lifted from C2≀C4
ρ134-400-202000040000-4000000    orthogonal lifted from C2≀C4
ρ1444440004000-100-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ154444000-4000-100-1-1-1-1-11111    orthogonal lifted from C2×F5
ρ16444-40000000-100-1-1-111-555-5    orthogonal lifted from C22⋊F5
ρ17444-40000000-100-1-1-1115-5-55    orthogonal lifted from C22⋊F5
ρ1844-400000000-10011-15-54ζ52+2ζ4ζ5443ζ54+2ζ43ζ524343ζ53+2ζ43ζ5434ζ54+2ζ4ζ534    orthogonal lifted from D10.D4
ρ1944-400000000-10011-15-54ζ54+2ζ4ζ53443ζ53+2ζ43ζ54343ζ54+2ζ43ζ52434ζ52+2ζ4ζ54    orthogonal lifted from D10.D4
ρ2044-400000000-10011-1-5543ζ53+2ζ43ζ5434ζ52+2ζ4ζ544ζ54+2ζ4ζ53443ζ54+2ζ43ζ5243    orthogonal lifted from D10.D4
ρ2144-400000000-10011-1-5543ζ54+2ζ43ζ52434ζ54+2ζ4ζ5344ζ52+2ζ4ζ5443ζ53+2ζ43ζ543    orthogonal lifted from D10.D4
ρ228-8000000000-200-25252000000    orthogonal faithful
ρ238-8000000000-20025-252000000    orthogonal faithful

Smallest permutation representation of C5⋊C2≀C4
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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 38)(22 37)(23 36)(24 40)(25 39)(26 33)(27 32)(28 31)(29 35)(30 34)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 22)(2 24 5 25)(3 21 4 23)(6 26 7 28)(8 30 10 29)(9 27)(11 36 17 33)(12 38 16 31)(13 40 20 34)(14 37 19 32)(15 39 18 35)

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

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

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

Matrix representation of C5⋊C2≀C4 in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1100
0000-1010
0000-1001
0000-1000
,
10000000
01000000
000-10000
00-100000
00000001
00000010
00000100
00001000
,
0-1000000
-10000000
000-10000
00-100000
00001000
00000100
00000010
00000001
,
10000000
01000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
00100000
00010000
10000000
0-1000000
0000-88130
0000580-8
0000-8085
00000138-8

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-8,5,-8,0,0,0,0,0,8,8,0,13,0,0,0,0,13,0,8,8,0,0,0,0,0,-8,5,-8] >;

C5⋊C2≀C4 in GAP, Magma, Sage, TeX

C_5\rtimes C_2\wr C_4
% in TeX

G:=Group("C5:C2wrC4");
// GroupNames label

G:=SmallGroup(320,202);
// by ID

G=gap.SmallGroup(320,202);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,675,570,297,1684,6278,3156]);
// Polycyclic

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

Export

Character table of C5⋊C2≀C4 in TeX

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