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G = C4×F5order 80 = 24·5

Direct product of C4 and F5

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

 Derived series C1 — C5 — C4×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C4×F5
 Lower central C5 — C4×F5
 Upper central C1 — C4

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

Character table of C4×F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 10 20A 20B size 1 1 5 5 1 1 5 5 5 5 5 5 5 5 5 5 4 4 4 4 ρ1 1 1 1 1 1 1 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 -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 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 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 -i i 1 -i i -i -i i 1 -1 -1 i 1 -1 i -i linear of order 4 ρ6 1 -1 -1 1 i -i i -i -1 1 -1 1 -i i -i i 1 -1 -i i linear of order 4 ρ7 1 -1 1 -1 i -i 1 i -i i i -i 1 -1 -1 -i 1 -1 -i i linear of order 4 ρ8 1 1 -1 -1 1 1 i -1 -i -i i i -i -i i -1 1 1 1 1 linear of order 4 ρ9 1 -1 -1 1 i -i -i -i 1 -1 1 -1 i -i i i 1 -1 -i i linear of order 4 ρ10 1 1 -1 -1 -1 -1 i 1 i i -i -i -i -i i 1 1 1 -1 -1 linear of order 4 ρ11 1 -1 -1 1 -i i i i 1 -1 1 -1 -i i -i -i 1 -1 i -i linear of order 4 ρ12 1 1 -1 -1 1 1 -i -1 i i -i -i i i -i -1 1 1 1 1 linear of order 4 ρ13 1 1 -1 -1 -1 -1 -i 1 -i -i i i i i -i 1 1 1 -1 -1 linear of order 4 ρ14 1 -1 1 -1 -i i -1 -i -i i i -i -1 1 1 i 1 -1 i -i linear of order 4 ρ15 1 -1 -1 1 -i i -i i -1 1 -1 1 i -i i -i 1 -1 i -i linear of order 4 ρ16 1 -1 1 -1 i -i -1 i i -i -i i -1 1 1 -i 1 -1 -i i linear of order 4 ρ17 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ18 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ19 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 -1 1 i -i complex faithful ρ20 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 -1 1 -i i 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)

 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 0 0 40 0 1 0 0 40 0 0 1 0 40 0 0 0 1 40
,
 40 0 0 0 0 0 0 0 40 0 0 40 0 0 0 0 0 0 0 40 0 0 40 0 0

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

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