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G = C6×F5order 120 = 23·3·5

Direct product of C6 and F5

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

Aliases: C6×F5, C10⋊C12, D5⋊C12, C302C4, D10.C6, C5⋊(C2×C12), C153(C2×C4), D5.(C2×C6), (C3×D5)⋊3C4, (C6×D5).3C2, (C3×D5).3C22, SmallGroup(120,40)

Series: Derived Chief Lower central Upper central

C1C5 — C6×F5
C1C5D5C3×D5C3×F5 — C6×F5
C5 — C6×F5
C1C6

Generators and relations for C6×F5
 G = < a,b,c | a6=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

5C2
5C2
5C4
5C22
5C4
5C6
5C6
5C2×C4
5C12
5C12
5C2×C6
5C2×C12

Character table of C6×F5

 class 12A2B2C3A3B4A4B4C4D56A6B6C6D6E6F1012A12B12C12D12E12F12G12H15A15B30A30B
 size 115511555541155554555555554444
ρ1111111111111111111111111111111    trivial
ρ21-1-11111-1-111-1-1-11-11-11-1-1-1-111111-1-1    linear of order 2
ρ31-1-1111-111-11-1-1-11-11-1-11111-1-1-111-1-1    linear of order 2
ρ4111111-1-1-1-111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ51111ζ3ζ32-1-1-1-11ζ32ζ3ζ3ζ32ζ32ζ31ζ6ζ65ζ65ζ6ζ6ζ65ζ65ζ6ζ3ζ32ζ32ζ3    linear of order 6
ρ61111ζ32ζ3-1-1-1-11ζ3ζ32ζ32ζ3ζ3ζ321ζ65ζ6ζ6ζ65ζ65ζ6ζ6ζ65ζ32ζ3ζ3ζ32    linear of order 6
ρ71111ζ3ζ3211111ζ32ζ3ζ3ζ32ζ32ζ31ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ81-1-11ζ32ζ31-1-111ζ65ζ6ζ6ζ3ζ65ζ32-1ζ3ζ6ζ6ζ65ζ65ζ32ζ32ζ3ζ32ζ3ζ65ζ6    linear of order 6
ρ91-1-11ζ3ζ321-1-111ζ6ζ65ζ65ζ32ζ6ζ3-1ζ32ζ65ζ65ζ6ζ6ζ3ζ3ζ32ζ3ζ32ζ6ζ65    linear of order 6
ρ101-1-11ζ3ζ32-111-11ζ6ζ65ζ65ζ32ζ6ζ3-1ζ6ζ3ζ3ζ32ζ32ζ65ζ65ζ6ζ3ζ32ζ6ζ65    linear of order 6
ρ111111ζ32ζ311111ζ3ζ32ζ32ζ3ζ3ζ321ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ121-1-11ζ32ζ3-111-11ζ65ζ6ζ6ζ3ζ65ζ32-1ζ65ζ32ζ32ζ3ζ3ζ6ζ6ζ65ζ32ζ3ζ65ζ6    linear of order 6
ρ1311-1-111-ii-ii111-1-1-1-11-ii-ii-ii-ii1111    linear of order 4
ρ1411-1-111i-ii-i111-1-1-1-11i-ii-ii-ii-i1111    linear of order 4
ρ151-11-111-i-iii1-1-11-11-1-1-i-ii-iii-ii11-1-1    linear of order 4
ρ161-11-111ii-i-i1-1-11-11-1-1ii-ii-i-ii-i11-1-1    linear of order 4
ρ171-11-1ζ3ζ32ii-i-i1ζ6ζ65ζ3ζ6ζ32ζ65-1ζ4ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ3ζ32ζ6ζ65    linear of order 12
ρ1811-1-1ζ32ζ3i-ii-i1ζ3ζ32ζ6ζ65ζ65ζ61ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ32ζ3ζ3ζ32    linear of order 12
ρ1911-1-1ζ32ζ3-ii-ii1ζ3ζ32ζ6ζ65ζ65ζ61ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ32ζ3ζ3ζ32    linear of order 12
ρ2011-1-1ζ3ζ32-ii-ii1ζ32ζ3ζ65ζ6ζ6ζ651ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ3ζ32ζ32ζ3    linear of order 12
ρ211-11-1ζ32ζ3-i-iii1ζ65ζ6ζ32ζ65ζ3ζ6-1ζ43ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ32ζ3ζ65ζ6    linear of order 12
ρ221-11-1ζ32ζ3ii-i-i1ζ65ζ6ζ32ζ65ζ3ζ6-1ζ4ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ32ζ3ζ65ζ6    linear of order 12
ρ2311-1-1ζ3ζ32i-ii-i1ζ32ζ3ζ65ζ6ζ6ζ651ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ3ζ32ζ32ζ3    linear of order 12
ρ241-11-1ζ3ζ32-i-iii1ζ6ζ65ζ3ζ6ζ32ζ65-1ζ43ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ3ζ32ζ6ζ65    linear of order 12
ρ254400440000-1440000-100000000-1-1-1-1    orthogonal lifted from F5
ρ264-400440000-1-4-40000100000000-1-111    orthogonal lifted from C2×F5
ρ274400-2+2-3-2-2-30000-1-2-2-3-2+2-30000-100000000ζ65ζ6ζ6ζ65    complex lifted from C3×F5
ρ284-400-2+2-3-2-2-30000-12+2-32-2-30000100000000ζ65ζ6ζ32ζ3    complex faithful
ρ294-400-2-2-3-2+2-30000-12-2-32+2-30000100000000ζ6ζ65ζ3ζ32    complex faithful
ρ304400-2-2-3-2+2-30000-1-2+2-3-2-2-30000-100000000ζ6ζ65ζ65ζ6    complex lifted from C3×F5

Permutation representations of C6×F5
On 30 points - transitive group 30T26
Generators in S30
(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)
(1 30 8 15 22)(2 25 9 16 23)(3 26 10 17 24)(4 27 11 18 19)(5 28 12 13 20)(6 29 7 14 21)
(1 4)(2 5)(3 6)(7 24 14 26)(8 19 15 27)(9 20 16 28)(10 21 17 29)(11 22 18 30)(12 23 13 25)

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

G:=Group( (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), (1,30,8,15,22)(2,25,9,16,23)(3,26,10,17,24)(4,27,11,18,19)(5,28,12,13,20)(6,29,7,14,21), (1,4)(2,5)(3,6)(7,24,14,26)(8,19,15,27)(9,20,16,28)(10,21,17,29)(11,22,18,30)(12,23,13,25) );

G=PermutationGroup([(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)], [(1,30,8,15,22),(2,25,9,16,23),(3,26,10,17,24),(4,27,11,18,19),(5,28,12,13,20),(6,29,7,14,21)], [(1,4),(2,5),(3,6),(7,24,14,26),(8,19,15,27),(9,20,16,28),(10,21,17,29),(11,22,18,30),(12,23,13,25)])

G:=TransitiveGroup(30,26);

C6×F5 is a maximal subgroup of   D6⋊F5  Dic3⋊F5

Matrix representation of C6×F5 in GL4(𝔽7) generated by

5000
0500
0050
0005
,
5645
1516
3614
4202
,
6040
0265
0534
0363
G:=sub<GL(4,GF(7))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[5,1,3,4,6,5,6,2,4,1,1,0,5,6,4,2],[6,0,0,0,0,2,5,3,4,6,3,6,0,5,4,3] >;

C6×F5 in GAP, Magma, Sage, TeX

C_6\times F_5
% in TeX

G:=Group("C6xF5");
// GroupNames label

G:=SmallGroup(120,40);
// by ID

G=gap.SmallGroup(120,40);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,60,1204,219]);
// Polycyclic

G:=Group<a,b,c|a^6=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 C6×F5 in TeX
Character table of C6×F5 in TeX

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