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G = C22⋊F5  order 80 = 24·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊F5, D10⋊2C4, 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 >

C1
C2
2C2
5C2
5C2
10C2
C5
C22
5C22
5C22
10C4
10C4
10C22
10C22
D5
C10
D5
2D5
2C10
5C23
5C2×C4
5C2×C4
C2×C10
D10
D10
2F5
2D10
2D10
2F5
5C22⋊C4
C2×F5
C22×D5
C2×F5
C22⋊F5

Character table of C22⋊F5

 class 12A2B2C2D2E4A4B4C4D510A10B10C
 size 1125510101010104444
ρ111111111111111    trivial
ρ211-111-1-11-111-1-11    linear of order 2
ρ311-111-11-11-11-1-11    linear of order 2
ρ4111111-1-1-1-11111    linear of order 2
ρ511-1-1-11i-i-ii1-1-11    linear of order 4
ρ6111-1-1-1-i-iii1111    linear of order 4
ρ7111-1-1-1ii-i-i1111    linear of order 4
ρ811-1-1-11-iii-i1-1-11    linear of order 4
ρ92-20-2200000200-2    orthogonal lifted from D4
ρ102-202-200000200-2    orthogonal lifted from D4
ρ1144-40000000-111-1    orthogonal lifted from C2×F5
ρ124440000000-1-1-1-1    orthogonal lifted from F5
ρ134-400000000-1√5-√51    orthogonal faithful
ρ144-400000000-1-√5√51    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  C102⋊4C4  C22⋊S5
C22⋊F5 is a maximal quotient of
 D10.D4  D10⋊C8  Dic5.D4  D10.3Q8  D20⋊C4  D4⋊F5  Q8⋊F5  Q8⋊2F5  C23⋊F5  C23.2F5  C23.F5  D6⋊F5  D10.D6  D25.D4  D5.D20  D10⋊F5  D10.D10  C102⋊C4  C102⋊4C4

Matrix representation of C22⋊F5 ►in GL6(𝔽41)

100000
0400000
001000
000100
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0004000
001600
00004035
0000635
,
010000
100000
000010
000001
0063500
00403500

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
 

Export

Subgroup lattice of C22⋊F5 in TeX
Character table of C22⋊F5 in TeX

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