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G = C22×C6  order 24 = 23·3

Abelian group of type [2,2,6]

direct product, abelian, monomial, 2-elementary

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 >

C1
C2
C2
C2
C2
C2
C2
C2
C3
C22
C22
C22
C22
C22
C22
C22
C6
C6
C6
C6
C6
C6
C6
C23
C2×C6
C2×C6
C2×C6
C2×C6
C2×C6
C2×C6
C2×C6
C22×C6

Character table of C22×C6

 class 12A2B2C2D2E2F2G3A3B6A6B6C6D6E6F6G6H6I6J6K6L6M6N
 size 111111111111111111111111
ρ1111111111111111111111111    trivial
ρ21-1111-1-1-111111-1-11-1-11-1-11-1-1    linear of order 2
ρ311-1-111-1-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ41-1-1-11-11111-1-11-1-11-1-1-111-111    linear of order 2
ρ5111-1-1-11-1111-1-111-1-1-1-1111-1-1    linear of order 2
ρ61-11-1-11-11111-1-1-1-1-111-1-1-1111    linear of order 2
ρ711-11-1-1-1111-11-111-1-1-11-1-1-111    linear of order 2
ρ81-1-11-111-111-11-1-1-1-111111-1-1-1    linear of order 2
ρ911111111ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ101-1111-1-1-1ζ32ζ3ζ32ζ32ζ3ζ65ζ6ζ32ζ65ζ6ζ3ζ65ζ6ζ3ζ65ζ6    linear of order 6
ρ1111-1-111-1-1ζ32ζ3ζ6ζ6ζ3ζ3ζ32ζ32ζ3ζ32ζ65ζ65ζ6ζ65ζ65ζ6    linear of order 6
ρ121-1-1-11-111ζ32ζ3ζ6ζ6ζ3ζ65ζ6ζ32ζ65ζ6ζ65ζ3ζ32ζ65ζ3ζ32    linear of order 6
ρ13111-1-1-11-1ζ32ζ3ζ32ζ6ζ65ζ3ζ32ζ6ζ65ζ6ζ65ζ3ζ32ζ3ζ65ζ6    linear of order 6
ρ141-11-1-11-11ζ32ζ3ζ32ζ6ζ65ζ65ζ6ζ6ζ3ζ32ζ65ζ65ζ6ζ3ζ3ζ32    linear of order 6
ρ1511-11-1-1-11ζ32ζ3ζ6ζ32ζ65ζ3ζ32ζ6ζ65ζ6ζ3ζ65ζ6ζ65ζ3ζ32    linear of order 6
ρ161-1-11-111-1ζ32ζ3ζ6ζ32ζ65ζ65ζ6ζ6ζ3ζ32ζ3ζ3ζ32ζ65ζ65ζ6    linear of order 6
ρ1711111111ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ181-1111-1-1-1ζ3ζ32ζ3ζ3ζ32ζ6ζ65ζ3ζ6ζ65ζ32ζ6ζ65ζ32ζ6ζ65    linear of order 6
ρ1911-1-111-1-1ζ3ζ32ζ65ζ65ζ32ζ32ζ3ζ3ζ32ζ3ζ6ζ6ζ65ζ6ζ6ζ65    linear of order 6
ρ201-1-1-11-111ζ3ζ32ζ65ζ65ζ32ζ6ζ65ζ3ζ6ζ65ζ6ζ32ζ3ζ6ζ32ζ3    linear of order 6
ρ21111-1-1-11-1ζ3ζ32ζ3ζ65ζ6ζ32ζ3ζ65ζ6ζ65ζ6ζ32ζ3ζ32ζ6ζ65    linear of order 6
ρ221-11-1-11-11ζ3ζ32ζ3ζ65ζ6ζ6ζ65ζ65ζ32ζ3ζ6ζ6ζ65ζ32ζ32ζ3    linear of order 6
ρ2311-11-1-1-11ζ3ζ32ζ65ζ3ζ6ζ32ζ3ζ65ζ6ζ65ζ32ζ6ζ65ζ6ζ32ζ3    linear of order 6
ρ241-1-11-111-1ζ3ζ32ζ65ζ3ζ6ζ6ζ65ζ65ζ32ζ3ζ32ζ32ζ3ζ6ζ6ζ65    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

100
060
006
,
600
010
001
,
300
020
005
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

Subgroup lattice of C22×C6 in TeX
Character table of C22×C6 in TeX

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