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## G = C2×C4×F5order 160 = 25·5

### Direct product of C2×C4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C4×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C2×C4×F5
 Lower central C5 — C2×C4×F5
 Upper central C1 — 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, bd=db, dcd-1=c3 >

Subgroups: 292 in 108 conjugacy classes, 62 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×F5, C2×C4×D5, C22×F5, C2×C4×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, F5, C2×C42, C2×F5, C4×F5, C22×F5, C2×C4×F5

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

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

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

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

C2×C4×F5 is a maximal subgroup of
C20.C42  D10.3M4(2)  M4(2)⋊3F5  C424F5  C10.(C4×D4)  C4⋊C45F5  C20⋊(C4⋊C4)  C2.(D4×F5)  (C2×F5)⋊Q8  D5⋊C4≀C2
C2×C4×F5 is a maximal quotient of
C42.5F5  C424F5  Dic5.C42  D10.C42  C20.12C42  M4(2)⋊5F5

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4X 5 10A 10B 10C 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 10 10 10 20 20 20 20 size 1 1 1 1 5 5 5 5 1 1 1 1 5 ··· 5 4 4 4 4 4 4 4 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 F5 C2×F5 C2×F5 C4×F5 kernel C2×C4×F5 C4×F5 C2×C4×D5 C22×F5 C4×D5 C2×Dic5 C2×C20 C2×F5 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 16 1 2 1 4

Matrix representation of C2×C4×F5 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 0 0 40 0 1 0 0 40 0 0 1 0 40 0 0 0 1 40
,
 9 0 0 0 0 0 0 0 9 0 0 9 0 0 0 0 0 0 0 9 0 0 9 0 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,0] >;

C2×C4×F5 in GAP, Magma, Sage, TeX

C_2\times C_4\times F_5
% in TeX

G:=Group("C2xC4xF5");
// GroupNames label

G:=SmallGroup(160,203);
// by ID

G=gap.SmallGroup(160,203);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,2309,599]);
// 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,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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