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G = C6.D6order 72 = 23·32

2nd non-split extension by C6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C6.2D6, Dic3Dic3, Dic32S3, C2.2S32, C3⋊S31C4, C31(C4×S3), C323(C2×C4), (C3×Dic3)⋊3C2, (C3×C6).2C22, (C2×C3⋊S3).1C2, SmallGroup(72,21)

Series: Derived Chief Lower central Upper central

C1C32 — C6.D6
C1C3C32C3×C6C3×Dic3 — C6.D6
C32 — C6.D6
C1C2

Generators and relations for C6.D6
 G = < a,b,c | a6=c2=1, b6=a3, bab-1=cac=a-1, cbc=b5 >

9C2
9C2
2C3
3C4
3C4
9C22
2C6
3S3
3S3
3S3
3S3
6S3
6S3
9C2×C4
3D6
3C12
3D6
3C12
6D6
3C4×S3
3C4×S3

Character table of C6.D6

 class 12A2B2C3A3B3C4A4B4C4D6A6B6C12A12B12C12D
 size 119922433332246666
ρ1111111111111111111    trivial
ρ211-1-11111-1-11111-11-11    linear of order 2
ρ311-1-1111-111-11111-11-1    linear of order 2
ρ41111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ51-11-1111-ii-ii-1-1-1-i-iii    linear of order 4
ρ61-1-11111-i-iii-1-1-1i-i-ii    linear of order 4
ρ71-11-1111i-ii-i-1-1-1ii-i-i    linear of order 4
ρ81-1-11111ii-i-i-1-1-1-iii-i    linear of order 4
ρ92200-12-10-2-202-1-11010    orthogonal lifted from D6
ρ1022002-1-1-200-2-12-10101    orthogonal lifted from D6
ρ1122002-1-12002-12-10-10-1    orthogonal lifted from S3
ρ122200-12-102202-1-1-10-10    orthogonal lifted from S3
ρ132-2002-1-12i00-2i1-210-i0i    complex lifted from C4×S3
ρ142-2002-1-1-2i002i1-210i0-i    complex lifted from C4×S3
ρ152-200-12-102i-2i0-211i0-i0    complex lifted from C4×S3
ρ162-200-12-10-2i2i0-211-i0i0    complex lifted from C4×S3
ρ174400-2-210000-2-210000    orthogonal lifted from S32
ρ184-400-2-21000022-10000    orthogonal faithful

Permutation representations of C6.D6
On 12 points - transitive group 12T39
Generators in S12
(1 3 5 7 9 11)(2 12 10 8 6 4)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 9)(3 7)(4 12)(6 10)

G:=sub<Sym(12)| (1,3,5,7,9,11)(2,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12), (1,9)(3,7)(4,12)(6,10)>;

G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12), (1,9)(3,7)(4,12)(6,10) );

G=PermutationGroup([(1,3,5,7,9,11),(2,12,10,8,6,4)], [(1,2,3,4,5,6,7,8,9,10,11,12)], [(1,9),(3,7),(4,12),(6,10)])

G:=TransitiveGroup(12,39);

On 24 points - transitive group 24T75
Generators in S24
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 16 18 20 22 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 24)(2 17)(3 22)(4 15)(5 20)(6 13)(7 18)(8 23)(9 16)(10 21)(11 14)(12 19)

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

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

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

G:=TransitiveGroup(24,75);

C6.D6 is a maximal subgroup of
S32⋊C4  C3⋊S3.Q8  Dic3.D6  D6.6D6  C4×S32  D6.3D6  Dic3⋊D6  C18.D6  C6.S32  C338(C2×C4)  C339(C2×C4)  Dic3.5S4  Dic32S4  C6.D30  Dic15⋊S3  C3⋊F5⋊S3
C6.D6 is a maximal quotient of
C12.29D6  C12.31D6  Dic32  C6.D12  C62.C22  C18.D6  He3⋊(C2×C4)  C338(C2×C4)  C339(C2×C4)  Dic32S4  C6.D30  Dic15⋊S3  C3⋊F5⋊S3

Polynomial with Galois group C6.D6 over ℚ
actionf(x)Disc(f)
12T39x12-8x10+24x8-32x6+17x4-8x2+8241·74·234

Matrix representation of C6.D6 in GL4(ℤ) generated by

1100
-1000
0011
00-10
,
00-1-1
0001
0-100
-1000
,
0-100
-1000
0011
000-1
G:=sub<GL(4,Integers())| [1,-1,0,0,1,0,0,0,0,0,1,-1,0,0,1,0],[0,0,0,-1,0,0,-1,0,-1,0,0,0,-1,1,0,0],[0,-1,0,0,-1,0,0,0,0,0,1,0,0,0,1,-1] >;

C6.D6 in GAP, Magma, Sage, TeX

C_6.D_6
% in TeX

G:=Group("C6.D6");
// GroupNames label

G:=SmallGroup(72,21);
// by ID

G=gap.SmallGroup(72,21);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,26,168,1204]);
// Polycyclic

G:=Group<a,b,c|a^6=c^2=1,b^6=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^5>;
// generators/relations

Export

Subgroup lattice of C6.D6 in TeX
Character table of C6.D6 in TeX

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