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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

C1C2 — C6×D4
C1C2C6C2×C6C3×D4 — C6×D4
C1C2 — C6×D4
C1C2×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 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×4], C22 [×4], C6, C6 [×2], C6 [×4], C2×C4, D4 [×4], C23 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C2×D4, C2×C12, C3×D4 [×4], C22×C6 [×2], C6×D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], C2×D4, C3×D4 [×2], C22×C6, C6×D4

Character table of C6×D4

 class 12A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N12A12B12C12D
 size 111122221122111111222222222222
ρ1111111111111111111111111111111    trivial
ρ211-1-1-111-111-11-11-1-11-1111-1-1-1-1111-1-1    linear of order 2
ρ311-1-1-11-11111-1-11-1-11-1-11-1-111-11-1-111    linear of order 2
ρ4111111-1-111-1-1111111-11-11-1-111-1-1-1-1    linear of order 2
ρ511-1-11-11-1111-1-11-1-11-11-111-1-11-1-1-111    linear of order 2
ρ61111-1-11111-1-11111111-11-111-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-1-11111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ811-1-11-1-1111-11-11-1-11-1-1-1-11111-111-1-1    linear of order 2
ρ91111-1-1-1-1ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ65ζ6ζ65ζ65ζ6ζ6ζ6ζ32ζ3ζ3ζ32    linear of order 6
ρ1011-1-1-11-11ζ3ζ321-1ζ65ζ32ζ6ζ65ζ3ζ6ζ65ζ3ζ6ζ65ζ3ζ32ζ6ζ32ζ6ζ65ζ3ζ32    linear of order 6
ρ1111-1-11-1-11ζ3ζ32-11ζ65ζ32ζ6ζ65ζ3ζ6ζ65ζ65ζ6ζ3ζ3ζ32ζ32ζ6ζ32ζ3ζ65ζ6    linear of order 6
ρ12111111-1-1ζ3ζ32-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ3ζ6ζ3ζ65ζ6ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 6
ρ1311-1-11-1-11ζ32ζ3-11ζ6ζ3ζ65ζ6ζ32ζ65ζ6ζ6ζ65ζ32ζ32ζ3ζ3ζ65ζ3ζ32ζ6ζ65    linear of order 6
ρ141111-1-111ζ32ζ3-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ6ζ3ζ6ζ32ζ3ζ65ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ1511-1-1-11-11ζ32ζ31-1ζ6ζ3ζ65ζ6ζ32ζ65ζ6ζ32ζ65ζ6ζ32ζ3ζ65ζ3ζ65ζ6ζ32ζ3    linear of order 6
ρ1611111111ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ1711-1-1-111-1ζ3ζ32-11ζ65ζ32ζ6ζ65ζ3ζ6ζ3ζ3ζ32ζ65ζ65ζ6ζ6ζ32ζ32ζ3ζ65ζ6    linear of order 6
ρ181111-1-111ζ3ζ32-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ65ζ32ζ65ζ3ζ32ζ6ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ1911111111ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ2011-1-11-11-1ζ3ζ321-1ζ65ζ32ζ6ζ65ζ3ζ6ζ3ζ65ζ32ζ3ζ65ζ6ζ32ζ6ζ6ζ65ζ3ζ32    linear of order 6
ρ2111-1-11-11-1ζ32ζ31-1ζ6ζ3ζ65ζ6ζ32ζ65ζ32ζ6ζ3ζ32ζ6ζ65ζ3ζ65ζ65ζ6ζ32ζ3    linear of order 6
ρ221111-1-1-1-1ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ6ζ65ζ6ζ6ζ65ζ65ζ65ζ3ζ32ζ32ζ3    linear of order 6
ρ2311-1-1-111-1ζ32ζ3-11ζ6ζ3ζ65ζ6ζ32ζ65ζ32ζ32ζ3ζ6ζ6ζ65ζ65ζ3ζ3ζ32ζ6ζ65    linear of order 6
ρ24111111-1-1ζ32ζ3-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ32ζ65ζ32ζ6ζ65ζ3ζ3ζ65ζ6ζ6ζ65    linear of order 6
ρ252-22-200002200-2-222-2-2000000000000    orthogonal lifted from D4
ρ262-2-22000022002-2-2-2-22000000000000    orthogonal lifted from D4
ρ272-22-20000-1--3-1+-3001+-31--3-1+-3-1--31+-31--3000000000000    complex lifted from C3×D4
ρ282-2-220000-1--3-1+-300-1--31--31--31+-31+-3-1+-3000000000000    complex lifted from C3×D4
ρ292-22-20000-1+-3-1--3001--31+-3-1--3-1+-31--31+-3000000000000    complex lifted from C3×D4
ρ302-2-220000-1+-3-1--300-1+-31+-31+-31--31--3-1--3000000000000    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);

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

400
0120
0012
,
100
01211
011
,
100
01211
001
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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