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G = C322Q8order 72 = 23·32

The semidirect product of C32 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C6.5D6, C322Q8, C31Dic6, Dic3.S3, C2.5S32, C3⋊Dic3.2C2, (C3×C6).5C22, (C3×Dic3).1C2, SmallGroup(72,24)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C322Q8
C1C3C32C3×C6C3×Dic3 — C322Q8
C32C3×C6 — C322Q8
C1C2

Generators and relations for C322Q8
 G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

2C3
3C4
3C4
9C4
2C6
9Q8
3Dic3
3C12
3C12
3Dic3
6Dic3
3Dic6
3Dic6

Character table of C322Q8

 class 123A3B3C4A4B4C6A6B6C12A12B12C12D
 size 1122466182246666
ρ1111111111111111    trivial
ρ211111-11-1111-111-1    linear of order 2
ρ311111-1-11111-1-1-1-1    linear of order 2
ρ4111111-1-11111-1-11    linear of order 2
ρ522-12-10202-1-10-1-10    orthogonal lifted from S3
ρ6222-1-1-200-12-11001    orthogonal lifted from D6
ρ7222-1-1200-12-1-100-1    orthogonal lifted from S3
ρ822-12-10-202-1-10110    orthogonal lifted from D6
ρ92-2222000-2-2-20000    symplectic lifted from Q8, Schur index 2
ρ102-22-1-10001-21-3003    symplectic lifted from Dic6, Schur index 2
ρ112-22-1-10001-21300-3    symplectic lifted from Dic6, Schur index 2
ρ122-2-12-1000-2110-330    symplectic lifted from Dic6, Schur index 2
ρ132-2-12-1000-21103-30    symplectic lifted from Dic6, Schur index 2
ρ1444-2-21000-2-210000    orthogonal lifted from S32
ρ154-4-2-2100022-10000    symplectic faithful, Schur index 2

Permutation representations of C322Q8
On 24 points - transitive group 24T62
Generators in S24
(1 19 14)(2 15 20)(3 17 16)(4 13 18)(5 21 10)(6 11 22)(7 23 12)(8 9 24)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 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 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

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

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

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

G:=TransitiveGroup(24,62);

C322Q8 is a maximal subgroup of
C322SD16  C32⋊Q16  S3×Dic6  Dic3.D6  D6.D6  D6.3D6  D6.4D6  C9⋊Dic6  He32Q8  C334Q8  C335Q8  CSU2(𝔽3)⋊S3  Dic3.S4  C3⋊Dic30  C323Dic10
C322Q8 is a maximal quotient of
Dic3⋊Dic3  C62.C22  C9⋊Dic6  He32Q8  C334Q8  C335Q8  Dic3.S4  C3⋊Dic30  C323Dic10

Matrix representation of C322Q8 in GL4(𝔽5) generated by

0010
0404
4040
0100
,
4040
0404
1000
0100
,
0400
1000
0004
0010
,
0303
3030
0002
0020
G:=sub<GL(4,GF(5))| [0,0,4,0,0,4,0,1,1,0,4,0,0,4,0,0],[4,0,1,0,0,4,0,1,4,0,0,0,0,4,0,0],[0,1,0,0,4,0,0,0,0,0,0,1,0,0,4,0],[0,3,0,0,3,0,0,0,0,3,0,2,3,0,2,0] >;

C322Q8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C3^2:2Q8");
// GroupNames label

G:=SmallGroup(72,24);
// by ID

G=gap.SmallGroup(72,24);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,168,1204]);
// Polycyclic

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

Export

Subgroup lattice of C322Q8 in TeX
Character table of C322Q8 in TeX

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