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## G = C2×C22⋊F5order 160 = 25·5

### Direct product of C2 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C22⋊F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C2×C22⋊F5
 Lower central C5 — C10 — C2×C22⋊F5
 Upper central C1 — C22 — C23

Generators and relations for C2×C22⋊F5
G = < a,b,c,d,e | a2=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 500 in 132 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C22, C22 [×2], C22 [×20], C5, C2×C4 [×8], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C22⋊C4 [×4], C22×C4 [×2], C24, F5 [×4], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4, C2×F5 [×4], C2×F5 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22⋊F5 [×4], C22×F5 [×2], C23×D5, C2×C22⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C22⋊F5 [×2], C22×F5, C2×C22⋊F5

Character table of C2×C22⋊F5

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5 10A 10B 10C 10D 10E 10F 10G size 1 1 1 1 2 2 5 5 5 5 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 1 -1 i 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 -1 -1 -i 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 -1 1 -i 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 1 1 i 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 1 1 -i -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 -1 1 i -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 -1 -1 i -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 1 -1 -i -i i -i i i -i i 1 -1 1 -1 -1 -1 1 1 linear of order 4 ρ17 2 2 -2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 -2 0 2 0 0 0 -2 orthogonal lifted from D4 ρ18 2 -2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 2 2 0 -2 0 0 0 -2 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 2 -2 0 2 0 0 0 -2 orthogonal lifted from D4 ρ20 2 -2 2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 0 -2 0 0 0 -2 orthogonal lifted from D4 ρ21 4 -4 -4 4 4 -4 0 0 0 0 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 4 4 0 0 0 0 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 -4 4 0 0 0 0 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 -4 -4 0 0 0 0 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 √5 -1 -√5 √5 -√5 1 orthogonal lifted from C22⋊F5 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 √5 1 √5 -√5 -√5 1 orthogonal lifted from C22⋊F5 ρ27 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -√5 1 -√5 √5 √5 1 orthogonal lifted from C22⋊F5 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -√5 -1 √5 -√5 √5 1 orthogonal lifted from C22⋊F5

Smallest permutation representation of C2×C22⋊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 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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 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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,2,3,4,5)(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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,2,3,4,5)(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)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(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)], [(1,2,3,4,5),(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×C22⋊F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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 39 0 0 0 0 0 1 0 0 0 0 0 0 22 38 0 3 0 0 0 19 38 3 0 0 3 38 19 0 0 0 3 0 38 22
,
 40 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 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 9 0 0 0 0 0 32 32 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;

C2×C22⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes F_5
% in TeX

G:=Group("C2xC2^2:F5");
// GroupNames label

G:=SmallGroup(160,212);
// by ID

G=gap.SmallGroup(160,212);
# by ID

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

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

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