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G = C2×Dic3order 24 = 23·3

Direct product of C2 and Dic3

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

Aliases: C2×Dic3, C6⋊C4, C2.2D6, C22.S3, C6.4C22, C32(C2×C4), (C2×C6).C2, SmallGroup(24,7)

Series: Derived Chief Lower central Upper central

C1C3 — C2×Dic3
C1C3C6Dic3 — C2×Dic3
C3 — C2×Dic3
C1C22

Generators and relations for C2×Dic3
 G = < a,b,c | a2=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3C2×C4

Character table of C2×Dic3

 class 12A2B2C34A4B4C4D6A6B6C
 size 111123333222
ρ1111111111111    trivial
ρ21-1-111-11-111-1-1    linear of order 2
ρ31-1-1111-11-11-1-1    linear of order 2
ρ411111-1-1-1-1111    linear of order 2
ρ51-11-11ii-i-i-1-11    linear of order 4
ρ611-1-11-iii-i-11-1    linear of order 4
ρ711-1-11i-i-ii-11-1    linear of order 4
ρ81-11-11-i-iii-1-11    linear of order 4
ρ92-2-22-10000-111    orthogonal lifted from D6
ρ102222-10000-1-1-1    orthogonal lifted from S3
ρ1122-2-2-100001-11    symplectic lifted from Dic3, Schur index 2
ρ122-22-2-1000011-1    symplectic lifted from Dic3, Schur index 2

Permutation representations of C2×Dic3
Regular action on 24 points - transitive group 24T6
Generators in S24
(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)
(1 16 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)

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

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

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

G:=TransitiveGroup(24,6);

Matrix representation of C2×Dic3 in GL3(𝔽13) generated by

1200
0120
0012
,
100
0112
010
,
100
058
008
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,1,1,0,12,0],[1,0,0,0,5,0,0,8,8] >;

C2×Dic3 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3
% in TeX

G:=Group("C2xDic3");
// GroupNames label

G:=SmallGroup(24,7);
// by ID

G=gap.SmallGroup(24,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,16,259]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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