Copied to
clipboard

G = C4×F5order 80 = 24·5

Direct product of C4 and F5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×F5, C5⋊C42, C202C4, Dic52C4, D10.4C22, D5.(C2×C4), C2.2(C2×F5), C10.3(C2×C4), (C4×D5).6C2, (C2×F5).2C2, SmallGroup(80,30)

Series: Derived Chief Lower central Upper central

C1C5 — C4×F5
C1C5D5D10C2×F5 — C4×F5
C5 — C4×F5
C1C4

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

5C2
5C2
5C4
5C4
5C22
5C4
5C4
5C4
5C2×C4
5C2×C4
5C2×C4
5C42

Character table of C4×F5

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L51020A20B
 size 11551155555555554444
ρ111111111111111111111    trivial
ρ2111111-11-1-1-1-1-1-1-111111    linear of order 2
ρ31111-1-11-1-1-1-1-1111-111-1-1    linear of order 2
ρ41111-1-1-1-11111-1-1-1-111-1-1    linear of order 2
ρ51-11-1-ii1-ii-i-ii1-1-1i1-1i-i    linear of order 4
ρ61-1-11i-ii-i-11-11-ii-ii1-1-ii    linear of order 4
ρ71-11-1i-i1i-iii-i1-1-1-i1-1-ii    linear of order 4
ρ811-1-111i-1-i-iii-i-ii-11111    linear of order 4
ρ91-1-11i-i-i-i1-11-1i-iii1-1-ii    linear of order 4
ρ1011-1-1-1-1i1ii-i-i-i-ii111-1-1    linear of order 4
ρ111-1-11-iiii1-11-1-ii-i-i1-1i-i    linear of order 4
ρ1211-1-111-i-1ii-i-iii-i-11111    linear of order 4
ρ1311-1-1-1-1-i1-i-iiiii-i111-1-1    linear of order 4
ρ141-11-1-ii-1-i-iii-i-111i1-1i-i    linear of order 4
ρ151-1-11-ii-ii-11-11i-ii-i1-1i-i    linear of order 4
ρ161-11-1i-i-1ii-i-ii-111-i1-1-ii    linear of order 4
ρ174400-4-40000000000-1-111    orthogonal lifted from C2×F5
ρ184400440000000000-1-1-1-1    orthogonal lifted from F5
ρ194-4004i-4i0000000000-11i-i    complex faithful
ρ204-400-4i4i0000000000-11-ii    complex faithful

Permutation representations of C4×F5
On 20 points - transitive group 20T20
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)
(1 11 6 16)(2 13 10 19)(3 15 9 17)(4 12 8 20)(5 14 7 18)

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

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

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

G:=TransitiveGroup(20,20);

C4×F5 is a maximal subgroup of   C8⋊F5  D4⋊F5  Q82F5  D10.C23  C523C42  GL2(𝔽5)
C4×F5 is a maximal quotient of   C8⋊F5  C10.C42  D10.3Q8  C523C42

Matrix representation of C4×F5 in GL5(𝔽41)

90000
040000
004000
000400
000040
,
10000
000040
010040
001040
000140
,
400000
000400
040000
000040
004000

G:=sub<GL(5,GF(41))| [9,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,40,0] >;

C4×F5 in GAP, Magma, Sage, TeX

C_4\times F_5
% in TeX

G:=Group("C4xF5");
// GroupNames label

G:=SmallGroup(80,30);
// by ID

G=gap.SmallGroup(80,30);
# by ID

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

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

Export

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

׿
×
𝔽