Copied to
clipboard

G = C6xD4  order 48 = 24·3

Direct product of C6 and D4

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

Aliases: C6xD4, C23:2C6, C12:4C22, C6.11C23, C4:(C2xC6), (C2xC4):2C6, (C2xC12):6C2, (C22xC6):1C2, (C2xC6):2C22, C22:2(C2xC6), C2.1(C22xC6), SmallGroup(48,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C6xD4
Chief series
C1 — C2 — C6 — C2xC6 — C3xD4 — C6xD4
Lower central
C1 — C2 — C6xD4
Upper central
C1 — C2xC6 — C6xD4

Generators and relations for C6xD4
 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, C2xC4, D4, C23, C12, C2xC6, C2xC6, C2xC6, C2xD4, C2xC12, C3xD4, C22xC6, C6xD4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C6xD4

Character table of C6xD4

 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 C3xD4
ρ282-2-220000-1-√-3-1+√-300-1-√-31-√-31-√-31+√-31+√-3-1+√-3000000000000    complex lifted from C3xD4
ρ292-22-20000-1+√-3-1-√-3001-√-31+√-3-1-√-3-1+√-31-√-31+√-3000000000000    complex lifted from C3xD4
ρ302-2-220000-1+√-3-1-√-300-1+√-31+√-31+√-31-√-31-√-3-1-√-3000000000000    complex lifted from C3xD4

Permutation representations of C6xD4
►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);
 

C6xD4 is a maximal subgroup of
 D4:Dic3  C12.D4  C23.7D6  D12:6C22  C23.23D6  C23.12D6  C23:2D6  D6:3D4  C23.14D6  C12:3D4  D4:6D6

Matrix representation of C6xD4 ►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] >;
 

C6xD4 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
 

Export

Character table of C6xD4 in TeX

׿
x
:
Z
F
o
wr
Q
<