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G = C4⋊F5order 80 = 24·5

The semidirect product of C4 and F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊F5, C201C4, D5.Q8, D5.1D4, Dic53C4, D10.5C22, C5⋊(C4⋊C4), (C2×F5).C2, C2.5(C2×F5), C10.4(C2×C4), (C4×D5).4C2, SmallGroup(80,31)

Series: Derived Chief Lower central Upper central

C1C10 — C4⋊F5
C1C5D5D10C2×F5 — C4⋊F5
C5C10 — C4⋊F5
C1C2C4

Generators and relations for C4⋊F5
 G = < a,b,c | a4=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

5C2
5C2
5C4
5C22
10C4
10C4
5C2×C4
5C2×C4
5C2×C4
2F5
2F5
5C4⋊C4

Character table of C4⋊F5

 class 12A2B2C4A4B4C4D4E4F51020A20B
 size 1155210101010104444
ρ111111111111111    trivial
ρ211111-1-1-1-111111    linear of order 2
ρ31111-1-111-1-111-1-1    linear of order 2
ρ41111-11-1-11-111-1-1    linear of order 2
ρ511-1-1-1ii-i-i111-1-1    linear of order 4
ρ611-1-11-ii-ii-11111    linear of order 4
ρ711-1-11i-ii-i-11111    linear of order 4
ρ811-1-1-1-i-iii111-1-1    linear of order 4
ρ92-22-20000002-200    orthogonal lifted from D4
ρ102-2-220000002-200    symplectic lifted from Q8, Schur index 2
ρ114400-400000-1-111    orthogonal lifted from C2×F5
ρ124400400000-1-1-1-1    orthogonal lifted from F5
ρ134-400000000-11--5-5    complex faithful
ρ144-400000000-11-5--5    complex faithful

Permutation representations of C4⋊F5
On 20 points - transitive group 20T18
Generators in S20
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
(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)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15), (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( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15), (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([(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)], [(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(20,18);

C4⋊F5 is a maximal subgroup of
C40⋊C4  D5.D8  D20⋊C4  Q8⋊F5  D10.C23  D4×F5  Q8×F5  Dic3⋊F5  C60⋊C4  C100⋊C4  D5.Dic10  Dic5⋊F5  C205F5  C20⋊F5  C202F5  C4⋊S5  A5⋊Q8
C4⋊F5 is a maximal quotient of
C40⋊C4  D5.D8  C40.C4  D10.Q8  C20⋊C8  Dic5⋊C8  D10.3Q8  Dic3⋊F5  C60⋊C4  C100⋊C4  D5.Dic10  Dic5⋊F5  C205F5  C20⋊F5  C202F5

Matrix representation of C4⋊F5 in GL4(𝔽3) generated by

1200
2200
2102
1010
,
0011
0202
2211
0212
,
1012
0001
0012
0111
G:=sub<GL(4,GF(3))| [1,2,2,1,2,2,1,0,0,0,0,1,0,0,2,0],[0,0,2,0,0,2,2,2,1,0,1,1,1,2,1,2],[1,0,0,0,0,0,0,1,1,0,1,1,2,1,2,1] >;

C4⋊F5 in GAP, Magma, Sage, TeX

C_4\rtimes F_5
% in TeX

G:=Group("C4:F5");
// GroupNames label

G:=SmallGroup(80,31);
// by ID

G=gap.SmallGroup(80,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,804,414]);
// Polycyclic

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

Export

Subgroup lattice of C4⋊F5 in TeX
Character table of C4⋊F5 in TeX

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