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G = C3×Q8  order 24 = 23·3

Direct product of C3 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×Q8, C4.C6, C12.3C2, C6.7C22, C2.2(C2×C6), SmallGroup(24,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C3×Q8
Chief series
C1 — C2 — C6 — C12 — C3×Q8
Lower central
C1 — C2 — C3×Q8
Upper central
C1 — C6 — C3×Q8

Generators and relations for C3×Q8
 G = < a,b,c | a3=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C3
C4
C4
C4
C6
Q8
C12
C12
C12
C3×Q8

Character table of C3×Q8

 class 123A3B4A4B4C6A6B12A12B12C12D12E12F
 size 111122211222222
ρ1111111111111111    trivial
ρ21111-11-111-111-1-1-1    linear of order 2
ρ31111-1-11111-1-1-11-1    linear of order 2
ρ411111-1-111-1-1-11-11    linear of order 2
ρ511ζ32ζ3111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ611ζ32ζ31-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ3    linear of order 6
ρ711ζ32ζ3-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ65    linear of order 6
ρ811ζ32ζ3-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ65    linear of order 6
ρ911ζ3ζ32-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ6    linear of order 6
ρ1011ζ3ζ321-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ32    linear of order 6
ρ1111ζ3ζ32-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ6    linear of order 6
ρ1211ζ3ζ32111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ132-222000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-2-1-√-3-1+√-30001+√-31-√-3000000    complex faithful
ρ152-2-1+√-3-1-√-30001-√-31+√-3000000    complex faithful

Permutation representations of C3×Q8
►Regular action on 24 points - transitive group 24T4
Generators in S24
(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 19 21)(6 20 22)(7 17 23)(8 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)

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

C3×Q8 is a maximal subgroup of
 Q8⋊2S3  C3⋊Q16  Q8⋊3S3  Q8⋊C9  Q8.A4  C4.F7  SU3(𝔽2)  Dic26⋊C3  Dic38⋊C3
C3×Q8 is a maximal quotient of
 C4.F7  Dic26⋊C3  Dic38⋊C3

Matrix representation of C3×Q8 ►in GL2(𝔽7) generated by

20
02
,
45
53
,
06
10
G:=sub<GL(2,GF(7))| [2,0,0,2],[4,5,5,3],[0,1,6,0] >;
 

C3×Q8 in GAP, Magma, Sage, TeX 

C_3\times Q_8
 
% in TeX
 
G:=Group("C3xQ8");
 
// GroupNames label
 
G:=SmallGroup(24,11);
 
// by ID
 
G=gap.SmallGroup(24,11);
 
# by ID
 
G:=PCGroup([4,-2,-2,-3,-2,48,113,53]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3×Q8 in TeX
Character table of C3×Q8 in TeX

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