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G = C4.F5order 80 = 24·5

The non-split extension by C4 of F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.F5, C20.1C4, C51M4(2), D10.3C4, Dic5.5C22, C5⋊C81C2, C2.4(C2×F5), C10.2(C2×C4), (C4×D5).3C2, SmallGroup(80,29)

Series: Derived Chief Lower central Upper central

C1C10 — C4.F5
C1C5C10Dic5C5⋊C8 — C4.F5
C5C10 — C4.F5
C1C2C4

Generators and relations for C4.F5
 G = < a,b,c | a4=b5=1, c4=a2, ab=ba, cac-1=a-1, cbc-1=b3 >

10C2
5C4
5C22
2D5
5C8
5C2×C4
5C8
5M4(2)

Character table of C4.F5

 class 12A2B4A4B4C58A8B8C8D1020A20B
 size 1110255410101010444
ρ111111111111111    trivial
ρ211-1-1111-11-111-1-1    linear of order 2
ρ311-1-11111-11-11-1-1    linear of order 2
ρ41111111-1-1-1-1111    linear of order 2
ρ5111-1-1-11i-i-ii1-1-1    linear of order 4
ρ611-11-1-11-i-iii111    linear of order 4
ρ711-11-1-11ii-i-i111    linear of order 4
ρ8111-1-1-11-iii-i1-1-1    linear of order 4
ρ92-200-2i2i20000-200    complex lifted from M4(2)
ρ102-2002i-2i20000-200    complex lifted from M4(2)
ρ11440-400-10000-111    orthogonal lifted from C2×F5
ρ12440400-10000-1-1-1    orthogonal lifted from F5
ρ134-40000-100001-5--5    complex faithful
ρ144-40000-100001--5-5    complex faithful

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

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

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

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

C4.F5 is a maximal subgroup of
C40.C4  D10.Q8  D4⋊F5  Q82F5  D5⋊M4(2)  D4.F5  Q8.F5  Dic3.F5  C12.F5  C100.C4  Dic5.F5  C524M4(2)  C20.12F5  C527M4(2)  C528M4(2)  C4.S5  C4.3S5
C4.F5 is a maximal quotient of
C20⋊C8  C10.C42  D10⋊C8  Dic3.F5  C12.F5  C100.C4  Dic5.F5  C524M4(2)  C20.12F5  C527M4(2)  C528M4(2)

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

2202
0121
2201
2010
,
1002
1001
1200
1111
,
0020
0011
2020
1101
G:=sub<GL(4,GF(3))| [2,0,2,2,2,1,2,0,0,2,0,1,2,1,1,0],[1,1,1,1,0,0,2,1,0,0,0,1,2,1,0,1],[0,0,2,1,0,0,0,1,2,1,2,0,0,1,0,1] >;

C4.F5 in GAP, Magma, Sage, TeX

C_4.F_5
% in TeX

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

G:=SmallGroup(80,29);
// by ID

G=gap.SmallGroup(80,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,42,804,414]);
// Polycyclic

G:=Group<a,b,c|a^4=b^5=1,c^4=a^2,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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