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## G = C22⋊F5order 80 = 24·5

### The semidirect product of C22 and F5 acting via F5/D5=C2

Aliases: C22⋊F5, D102C4, D5.2D4, D10.6C22, (C2×F5)⋊C2, C5⋊(C22⋊C4), (C2×C10)⋊1C4, C2.7(C2×F5), C10.7(C2×C4), (C22×D5).2C2, SmallGroup(80,34)

Series: Derived Chief Lower central Upper central

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

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

Character table of C22⋊F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5 10A 10B 10C size 1 1 2 5 5 10 10 10 10 10 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 -1 -1 1 i -i -i i 1 -1 -1 1 linear of order 4 ρ6 1 1 1 -1 -1 -1 -i -i i i 1 1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 i i -i -i 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 -1 1 -i i i -i 1 -1 -1 1 linear of order 4 ρ9 2 -2 0 -2 2 0 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ10 2 -2 0 2 -2 0 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F5 ρ12 4 4 4 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ13 4 -4 0 0 0 0 0 0 0 0 -1 √5 -√5 1 orthogonal faithful ρ14 4 -4 0 0 0 0 0 0 0 0 -1 -√5 √5 1 orthogonal faithful

Permutation representations of C22⋊F5
On 20 points - transitive group 20T19
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(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)(11 16)(12 18 15 19)(13 20 14 17)

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

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

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

G:=TransitiveGroup(20,19);

On 20 points - transitive group 20T22
Generators in S20
(11 16)(12 17)(13 18)(14 19)(15 20)
(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)
(1 18 6 13)(2 20 10 11)(3 17 9 14)(4 19 8 12)(5 16 7 15)

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

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

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

G:=TransitiveGroup(20,22);

C22⋊F5 is a maximal subgroup of
D10.D4  C23⋊F5  D10.C23  D4×F5  D6⋊F5  D10.D6  A4⋊F5  D25.D4  D5.D20  D10⋊F5  D10.D10  C102⋊C4  C1024C4  C22⋊S5
C22⋊F5 is a maximal quotient of
D10.D4  D10⋊C8  Dic5.D4  D10.3Q8  D20⋊C4  D4⋊F5  Q8⋊F5  Q82F5  C23⋊F5  C23.2F5  C23.F5  D6⋊F5  D10.D6  D25.D4  D5.D20  D10⋊F5  D10.D10  C102⋊C4  C1024C4

Matrix representation of C22⋊F5 in GL6(𝔽41)

 1 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 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 6 0 0 0 0 0 0 40 35 0 0 0 0 6 35
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 6 35 0 0 0 0 40 35 0 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,1,0,0,0,0,0,0,1,0,0] >;

C22⋊F5 in GAP, Magma, Sage, TeX

C_2^2\rtimes F_5
% in TeX

G:=Group("C2^2:F5");
// GroupNames label

G:=SmallGroup(80,34);
// by ID

G=gap.SmallGroup(80,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,804,414]);
// Polycyclic

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

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