Copied to
clipboard

## G = C22×C6order 24 = 23·3

### Abelian group of type [2,2,6]

Aliases: C22×C6, SmallGroup(24,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6
 Lower central C1 — C22×C6
 Upper central C1 — C22×C6

Generators and relations for C22×C6
G = < a,b,c | a2=b2=c6=1, ab=ba, ac=ca, bc=cb >

Character table of C22×C6

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 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 -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 -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 1 -1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ6 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 linear of order 2 ρ7 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ8 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ10 1 -1 1 1 1 -1 -1 -1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 ζ3 ζ65 ζ6 ζ3 ζ65 ζ6 linear of order 6 ρ11 1 1 -1 -1 1 1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ65 ζ65 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ12 1 -1 -1 -1 1 -1 1 1 ζ32 ζ3 ζ6 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 ζ32 ζ65 ζ3 ζ32 linear of order 6 ρ13 1 1 1 -1 -1 -1 1 -1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ3 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ14 1 -1 1 -1 -1 1 -1 1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 linear of order 6 ρ15 1 1 -1 1 -1 -1 -1 1 ζ32 ζ3 ζ6 ζ32 ζ65 ζ3 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ16 1 -1 -1 1 -1 1 1 -1 ζ32 ζ3 ζ6 ζ32 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 linear of order 6 ρ17 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ18 1 -1 1 1 1 -1 -1 -1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 ζ32 ζ6 ζ65 ζ32 ζ6 ζ65 linear of order 6 ρ19 1 1 -1 -1 1 1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ6 ζ6 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ20 1 -1 -1 -1 1 -1 1 1 ζ3 ζ32 ζ65 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 ζ3 ζ6 ζ32 ζ3 linear of order 6 ρ21 1 1 1 -1 -1 -1 1 -1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ32 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ22 1 -1 1 -1 -1 1 -1 1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 linear of order 6 ρ23 1 1 -1 1 -1 -1 -1 1 ζ3 ζ32 ζ65 ζ3 ζ6 ζ32 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ24 1 -1 -1 1 -1 1 1 -1 ζ3 ζ32 ζ65 ζ3 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 linear of order 6

Permutation representations of C22×C6
Regular action on 24 points - transitive group 24T3
Generators in S24
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,3);

C22×C6 is a maximal subgroup of   C6.D4

Matrix representation of C22×C6 in GL3(𝔽7) generated by

 1 0 0 0 6 0 0 0 6
,
 6 0 0 0 1 0 0 0 1
,
 3 0 0 0 2 0 0 0 5
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[6,0,0,0,1,0,0,0,1],[3,0,0,0,2,0,0,0,5] >;

C22×C6 in GAP, Magma, Sage, TeX

C_2^2\times C_6
% in TeX

G:=Group("C2^2xC6");
// GroupNames label

G:=SmallGroup(24,15);
// by ID

G=gap.SmallGroup(24,15);
# by ID

G:=PCGroup([4,-2,-2,-2,-3]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

׿
×
𝔽