Copied to
clipboard

## G = C2×D5⋊F5order 400 = 24·52

### Direct product of C2 and D5⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C5⋊F5 — D5⋊F5 — C2×D5⋊F5
 Lower central C52 — C2×D5⋊F5
 Upper central C1 — C2

Generators and relations for C2×D5⋊F5
G = < a,b,c,d,e | a2=b5=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 892 in 124 conjugacy classes, 39 normal (11 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C5 [×2], C5 [×2], C2×C4 [×6], C23, D5 [×4], D5 [×8], C10 [×2], C10 [×6], C22×C4, F5 [×12], D10 [×2], D10 [×12], C2×C10 [×2], C52, C2×F5 [×14], C22×D5 [×2], C5×D5 [×4], C5⋊D5 [×2], C5×C10, C22×F5 [×2], C5⋊F5 [×2], C52⋊C4 [×2], D52 [×4], D5×C10 [×2], C2×C5⋊D5, D5⋊F5 [×4], C2×C5⋊F5, C2×C52⋊C4, C2×D52, C2×D5⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, F5 [×2], C2×F5 [×6], C22×F5 [×2], D5⋊F5, C2×D5⋊F5

Character table of C2×D5⋊F5

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H size 1 1 5 5 5 5 25 25 25 25 25 25 25 25 25 25 4 4 8 8 4 4 8 8 20 20 20 20 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 linear of order 2 ρ5 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 linear of order 2 ρ6 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 linear of order 2 ρ9 1 1 1 -1 -1 1 -1 -1 -i -i i -i i i -i i 1 1 1 1 1 1 1 1 -1 1 -1 1 linear of order 4 ρ10 1 -1 -1 1 -1 1 -1 1 i i -i -i i i -i -i 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 4 ρ11 1 -1 -1 1 -1 1 -1 1 -i -i i i -i -i i i 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 4 ρ12 1 1 1 -1 -1 1 -1 -1 i i -i i -i -i i -i 1 1 1 1 1 1 1 1 -1 1 -1 1 linear of order 4 ρ13 1 1 -1 1 1 -1 -1 -1 -i i -i i -i i -i i 1 1 1 1 1 1 1 1 1 -1 1 -1 linear of order 4 ρ14 1 -1 1 -1 1 -1 -1 1 i -i i i -i i -i -i 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 4 ρ15 1 -1 1 -1 1 -1 -1 1 -i i -i -i i -i i i 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 4 ρ16 1 1 -1 1 1 -1 -1 -1 i -i i -i i -i i -i 1 1 1 1 1 1 1 1 1 -1 1 -1 linear of order 4 ρ17 4 -4 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 -1 0 1 0 orthogonal lifted from C2×F5 ρ18 4 -4 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 0 1 0 -1 orthogonal lifted from C2×F5 ρ19 4 4 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 1 0 1 0 orthogonal lifted from C2×F5 ρ20 4 -4 4 0 0 -4 0 0 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 0 -1 0 1 orthogonal lifted from C2×F5 ρ21 4 4 0 4 4 0 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 -1 0 -1 0 orthogonal lifted from F5 ρ22 4 -4 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 1 0 -1 0 orthogonal lifted from C2×F5 ρ23 4 4 4 0 0 4 0 0 0 0 0 0 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 0 -1 0 -1 orthogonal lifted from F5 ρ24 4 4 -4 0 0 -4 0 0 0 0 0 0 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 0 1 0 1 orthogonal lifted from C2×F5 ρ25 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 3 -2 2 2 -3 2 0 0 0 0 orthogonal faithful ρ26 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 3 -2 -2 -2 3 -2 0 0 0 0 orthogonal lifted from D5⋊F5 ρ27 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 3 -2 -2 -2 3 0 0 0 0 orthogonal lifted from D5⋊F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 3 2 2 2 -3 0 0 0 0 orthogonal faithful

Permutation representations of C2×D5⋊F5
On 20 points - transitive group 20T114
Generators in S20
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 13)(7 12)(8 11)(9 15)(10 14)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(6 9 10 7)(11 15 13 14)(16 20 18 19)

G:=sub<Sym(20)| (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(6,9,10,7)(11,15,13,14)(16,20,18,19)>;

G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(6,9,10,7)(11,15,13,14)(16,20,18,19) );

G=PermutationGroup([(1,8),(2,9),(3,10),(4,6),(5,7),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,13),(7,12),(8,11),(9,15),(10,14)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(2,3,5,4),(6,9,10,7),(11,15,13,14),(16,20,18,19)])

G:=TransitiveGroup(20,114);

Matrix representation of C2×D5⋊F5 in GL8(ℤ)

 -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 0 1 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 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 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 1 1 1 1 0 0 0 0 -1 0 0 0 0 0 0 -1 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 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0
,
 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 1 1

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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0],[0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1] >;

C2×D5⋊F5 in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes F_5
% in TeX

G:=Group("C2xD5:F5");
// GroupNames label

G:=SmallGroup(400,210);
// by ID

G=gap.SmallGroup(400,210);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,1444,970,262,8645,1463]);
// Polycyclic

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

Export

׿
×
𝔽