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## G = C6×D4order 48 = 24·3

### Direct product of C6 and D4

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

Aliases: C6×D4, C232C6, C124C22, C6.11C23, C4⋊(C2×C6), (C2×C4)⋊2C6, (C2×C12)⋊6C2, (C22×C6)⋊1C2, (C2×C6)⋊2C22, C222(C2×C6), C2.1(C22×C6), SmallGroup(48,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×D4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4
 Lower central C1 — C2 — C6×D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C6×D4
G = < a,b,c | a6=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C12, C2×C6, C2×C6, C2×C6, C2×D4, C2×C12, C3×D4, C22×C6, C6×D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C6×D4

Character table of C6×D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 12A 12B 12C 12D size 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 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 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 -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 -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 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 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 -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 -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 1 -1 1 1 -1 -1 linear of order 2 ρ9 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ6 ζ32 ζ3 ζ3 ζ32 linear of order 6 ρ10 1 1 -1 -1 -1 1 -1 1 ζ3 ζ32 1 -1 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 ζ3 ζ6 ζ65 ζ3 ζ32 ζ6 ζ32 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ11 1 1 -1 -1 1 -1 -1 1 ζ3 ζ32 -1 1 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ12 1 1 1 1 1 1 -1 -1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ65 ζ3 ζ6 ζ3 ζ65 ζ6 ζ32 ζ32 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ13 1 1 -1 -1 1 -1 -1 1 ζ32 ζ3 -1 1 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ14 1 1 1 1 -1 -1 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ6 ζ3 ζ6 ζ32 ζ3 ζ65 ζ65 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ15 1 1 -1 -1 -1 1 -1 1 ζ32 ζ3 1 -1 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 ζ32 ζ65 ζ6 ζ32 ζ3 ζ65 ζ3 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ16 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ17 1 1 -1 -1 -1 1 1 -1 ζ3 ζ32 -1 1 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ18 1 1 1 1 -1 -1 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ65 ζ32 ζ65 ζ3 ζ32 ζ6 ζ6 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ19 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ20 1 1 -1 -1 1 -1 1 -1 ζ3 ζ32 1 -1 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ3 ζ65 ζ32 ζ3 ζ65 ζ6 ζ32 ζ6 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ21 1 1 -1 -1 1 -1 1 -1 ζ32 ζ3 1 -1 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ32 ζ6 ζ3 ζ32 ζ6 ζ65 ζ3 ζ65 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ22 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ65 ζ3 ζ32 ζ32 ζ3 linear of order 6 ρ23 1 1 -1 -1 -1 1 1 -1 ζ32 ζ3 -1 1 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ24 1 1 1 1 1 1 -1 -1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ6 ζ32 ζ65 ζ32 ζ6 ζ65 ζ3 ζ3 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ25 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 -2 -2 2 0 0 0 0 2 2 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 -2 2 -2 0 0 0 0 -1-√-3 -1+√-3 0 0 1+√-3 1-√-3 -1+√-3 -1-√-3 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ28 2 -2 -2 2 0 0 0 0 -1-√-3 -1+√-3 0 0 -1-√-3 1-√-3 1-√-3 1+√-3 1+√-3 -1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ29 2 -2 2 -2 0 0 0 0 -1+√-3 -1-√-3 0 0 1-√-3 1+√-3 -1-√-3 -1+√-3 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ30 2 -2 -2 2 0 0 0 0 -1+√-3 -1-√-3 0 0 -1+√-3 1+√-3 1+√-3 1-√-3 1-√-3 -1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×D4

Permutation representations of C6×D4
On 24 points - transitive group 24T38
Generators in S24
(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 8 14 19)(2 9 15 20)(3 10 16 21)(4 11 17 22)(5 12 18 23)(6 7 13 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(24,38);

C6×D4 is a maximal subgroup of
D4⋊Dic3  C12.D4  C23.7D6  D126C22  C23.23D6  C23.12D6  C232D6  D63D4  C23.14D6  C123D4  D46D6

Matrix representation of C6×D4 in GL3(𝔽13) generated by

 4 0 0 0 12 0 0 0 12
,
 1 0 0 0 12 11 0 1 1
,
 1 0 0 0 12 11 0 0 1
G:=sub<GL(3,GF(13))| [4,0,0,0,12,0,0,0,12],[1,0,0,0,12,1,0,11,1],[1,0,0,0,12,0,0,11,1] >;

C6×D4 in GAP, Magma, Sage, TeX

C_6\times D_4
% in TeX

G:=Group("C6xD4");
// GroupNames label

G:=SmallGroup(48,45);
// by ID

G=gap.SmallGroup(48,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-2,261]);
// Polycyclic

G:=Group<a,b,c|a^6=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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