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## G = C22×F5order 80 = 24·5

### Direct product of C22 and F5

Aliases: C22×F5, D103C4, D5.C23, D10.7C22, C10⋊(C2×C4), D5⋊(C2×C4), C5⋊(C22×C4), (C2×C10)⋊2C4, (C22×D5).3C2, SmallGroup(80,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×F5
 Chief series C1 — C5 — D5 — F5 — C2×F5 — C22×F5
 Lower central C5 — C22×F5
 Upper central C1 — C22

Generators and relations for C22×F5
G = < a,b,c,d | a2=b2=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 142 in 54 conjugacy classes, 32 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C22, C22 [×6], C5, C2×C4 [×6], C23, D5, D5 [×3], C10 [×3], C22×C4, F5 [×4], D10 [×6], C2×C10, C2×F5 [×6], C22×D5, C22×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, F5, C2×F5 [×3], C22×F5

Character table of C22×F5

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5 10A 10B 10C size 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 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 trivial ρ2 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 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 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 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 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 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 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 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 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 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 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 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 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 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 linear of order 4 ρ17 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ18 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ20 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×F5

Permutation representations of C22×F5
On 20 points - transitive group 20T16
Generators in S20
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 6)(2 7)(3 8)(4 9)(5 10)(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)

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

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

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

G:=TransitiveGroup(20,16);

C22×F5 is a maximal subgroup of   D10.3Q8
C22×F5 is a maximal quotient of   D5⋊M4(2)  D10.C23  D4.F5  Q8.F5

Matrix representation of C22×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 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 40 40 40 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 32 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 40 0 0 0 1 1 1 1

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],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,40,0,0,0],[32,0,0,0,0,0,40,0,0,1,0,0,0,40,1,0,0,0,0,1,0,0,40,0,1] >;

C22×F5 in GAP, Magma, Sage, TeX

C_2^2\times F_5
% in TeX

G:=Group("C2^2xF5");
// GroupNames label

G:=SmallGroup(80,50);
// by ID

G=gap.SmallGroup(80,50);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,804,219]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=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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