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G = C6.D4order 48 = 24·3

7th non-split extension by C6 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.11D4, C23.2S3, C22.7D6, C222Dic3, (C2×C6)⋊2C4, C6.9(C2×C4), C32(C22⋊C4), (C2×Dic3)⋊2C2, C2.3(C3⋊D4), (C22×C6).2C2, (C2×C6).7C22, C2.5(C2×Dic3), SmallGroup(48,19)

Series: Derived Chief Lower central Upper central

C1C6 — C6.D4
C1C3C6C2×C6C2×Dic3 — C6.D4
C3C6 — C6.D4
C1C22C23

Generators and relations for C6.D4
 G = < a,b,c | a6=b4=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >

2C2
2C2
2C22
2C22
6C4
6C4
2C6
2C6
3C2×C4
3C2×C4
2Dic3
2C2×C6
2C2×C6
2Dic3
3C22⋊C4

Character table of C6.D4

 class 12A2B2C2D2E34A4B4C4D6A6B6C6D6E6F6G
 size 111122266662222222
ρ1111111111111111111    trivial
ρ21111-1-11-11-111-1-111-1-1    linear of order 2
ρ31111111-1-1-1-11111111    linear of order 2
ρ41111-1-111-11-11-1-111-1-1    linear of order 2
ρ51-11-11-11-i-iii1-1-1-1-111    linear of order 4
ρ61-11-1-111i-i-ii111-1-1-1-1    linear of order 4
ρ71-11-1-111-iii-i111-1-1-1-1    linear of order 4
ρ81-11-11-11ii-i-i1-1-1-1-111    linear of order 4
ρ9222222-10000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-2-2-10000-111-1-111    orthogonal lifted from D6
ρ112-2-220020000-2002-200    orthogonal lifted from D4
ρ1222-2-20020000-200-2200    orthogonal lifted from D4
ρ132-22-2-22-10000-1-1-11111    symplectic lifted from Dic3, Schur index 2
ρ142-22-22-2-10000-11111-1-1    symplectic lifted from Dic3, Schur index 2
ρ152-2-2200-100001--3-3-11-3--3    complex lifted from C3⋊D4
ρ1622-2-200-100001-3--31-1-3--3    complex lifted from C3⋊D4
ρ172-2-2200-100001-3--3-11--3-3    complex lifted from C3⋊D4
ρ1822-2-200-100001--3-31-1--3-3    complex lifted from C3⋊D4

Permutation representations of C6.D4
On 24 points - transitive group 24T44
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 23 18 10)(2 22 13 9)(3 21 14 8)(4 20 15 7)(5 19 16 12)(6 24 17 11)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 15 10 18)(8 14 11 17)(9 13 12 16)

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

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

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

G:=TransitiveGroup(24,44);

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

1200
040
0010
,
500
001
010
,
800
001
0120
G:=sub<GL(3,GF(13))| [12,0,0,0,4,0,0,0,10],[5,0,0,0,0,1,0,1,0],[8,0,0,0,0,12,0,1,0] >;

C6.D4 in GAP, Magma, Sage, TeX

C_6.D_4
% in TeX

G:=Group("C6.D4");
// GroupNames label

G:=SmallGroup(48,19);
// by ID

G=gap.SmallGroup(48,19);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,804]);
// Polycyclic

G:=Group<a,b,c|a^6=b^4=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations

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