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## G = C2×C4⋊F5order 160 = 25·5

### Direct product of C2 and C4⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4⋊F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C2×C4⋊F5
 Lower central C5 — C10 — C2×C4⋊F5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C4⋊F5
G = < a,b,c,d | a2=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 292 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×C4⋊C4, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C4⋊F5 [×4], C2×C4×D5, C22×F5 [×2], C2×C4⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×F5 [×3], C4⋊F5 [×2], C22×F5, C2×C4⋊F5

Character table of C2×C4⋊F5

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 5 5 5 5 2 2 10 10 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 4 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 -1 1 i -1 1 i -i i -i -i i -i 1 -1 1 -1 -1 1 -1 1 linear of order 4 ρ10 1 1 -1 -1 1 -1 -1 1 1 -1 -i 1 -1 i -i -i i -i i i 1 -1 1 -1 1 -1 1 -1 linear of order 4 ρ11 1 1 1 1 -1 -1 -1 -1 1 1 -i -1 -1 i -i i -i i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 -1 -1 i 1 1 i -i -i i i -i -i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ13 1 1 1 1 -1 -1 -1 -1 -1 -1 -i 1 1 -i i i -i -i i i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 1 1 i -1 -1 -i i -i i -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ15 1 1 -1 -1 1 -1 -1 1 1 -1 i 1 -1 -i i i -i i -i -i 1 -1 1 -1 1 -1 1 -1 linear of order 4 ρ16 1 1 -1 -1 1 -1 -1 1 -1 1 -i -1 1 -i i -i i i -i i 1 -1 1 -1 -1 1 -1 1 linear of order 4 ρ17 2 -2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 4 -4 -4 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 -1 1 -1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 4 4 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ24 4 4 -4 -4 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 1 -1 1 -1 orthogonal lifted from C2×F5 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4⋊F5 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4⋊F5 ρ27 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 √-5 -√-5 -√-5 √-5 complex lifted from C4⋊F5 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 -√-5 √-5 √-5 -√-5 complex lifted from C4⋊F5

Smallest permutation representation of C2×C4⋊F5
On 40 points
Generators in S40
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 21 16 26)(12 22 17 27)(13 23 18 28)(14 24 19 29)(15 25 20 30)
(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 3 5 4)(6 8 7 10)(11 18 12 20)(13 17 15 16)(14 19)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)

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

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

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

Matrix representation of C2×C4⋊F5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 35 5 0 0 0 0 9 6 0 0 0 0 0 0 34 0 27 27 0 0 14 7 14 0 0 0 0 14 7 14 0 0 27 27 0 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 40 40 40
,
 9 0 0 0 0 0 38 32 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,9,0,0,0,0,5,6,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[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],[9,38,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C2×C4⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes F_5
% in TeX

G:=Group("C2xC4:F5");
// GroupNames label

G:=SmallGroup(160,204);
// by ID

G=gap.SmallGroup(160,204);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,2309,599]);
// Polycyclic

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

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