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## G = C32⋊2SD16order 144 = 24·32

### The semidirect product of C32 and SD16 acting via SD16/C2=D4

Aliases: C322SD16, C2.4S3≀C2, (C3×C6).4D4, D6⋊S3.C2, C322C82C2, C322Q81C2, C3⋊Dic3.2C22, SmallGroup(144,118)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C32⋊2SD16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊2SD16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C32⋊2SD16
 Upper central C1 — C2

Generators and relations for C322SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=b-1, dad=a-1, cbc-1=a, bd=db, dcd=c3 >

Character table of C322SD16

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 12A 12B size 1 1 12 4 4 12 18 4 4 12 12 18 18 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 2 2 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 2 2 0 0 -2 -2 0 0 √-2 -√-2 0 0 complex lifted from SD16 ρ7 2 -2 0 2 2 0 0 -2 -2 0 0 -√-2 √-2 0 0 complex lifted from SD16 ρ8 4 4 0 -2 1 2 0 1 -2 0 0 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ9 4 4 2 1 -2 0 0 -2 1 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ10 4 4 0 -2 1 -2 0 1 -2 0 0 0 0 1 1 orthogonal lifted from S3≀C2 ρ11 4 4 -2 1 -2 0 0 -2 1 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ12 4 -4 0 -2 1 0 0 -1 2 0 0 0 0 √3 -√3 symplectic faithful, Schur index 2 ρ13 4 -4 0 -2 1 0 0 -1 2 0 0 0 0 -√3 √3 symplectic faithful, Schur index 2 ρ14 4 -4 0 1 -2 0 0 2 -1 -√-3 √-3 0 0 0 0 complex faithful ρ15 4 -4 0 1 -2 0 0 2 -1 √-3 -√-3 0 0 0 0 complex faithful

Permutation representations of C322SD16
On 24 points - transitive group 24T217
Generators in S24
(2 13 22)(4 24 15)(6 9 18)(8 20 11)
(1 21 12)(3 14 23)(5 17 16)(7 10 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,217);

On 24 points - transitive group 24T220
Generators in S24
(1 12 21)(3 23 14)(5 16 17)(7 19 10)
(2 13 22)(4 24 15)(6 9 18)(8 20 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 20)(10 23)(11 18)(12 21)(13 24)(14 19)(15 22)(16 17)

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

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

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

G:=TransitiveGroup(24,220);

C322SD16 is a maximal subgroup of
C32⋊D85C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  C3⋊S32SD16  C62.12D4  C62.13D4  C62.15D4  C336SD16  C337SD16  C338SD16
C322SD16 is a maximal quotient of
C62.3D4  C62.4D4  C62.6D4  He32SD16  C336SD16  C337SD16  C338SD16

Matrix representation of C322SD16 in GL4(𝔽7) generated by

 3 0 6 0 4 6 3 6 1 1 5 4 1 0 3 1
,
 5 1 2 0 0 5 1 4 1 2 5 0 5 2 4 0
,
 5 6 4 3 4 0 6 1 5 5 1 2 2 2 3 1
,
 6 6 5 6 4 6 3 6 5 4 2 1 0 1 2 0
G:=sub<GL(4,GF(7))| [3,4,1,1,0,6,1,0,6,3,5,3,0,6,4,1],[5,0,1,5,1,5,2,2,2,1,5,4,0,4,0,0],[5,4,5,2,6,0,5,2,4,6,1,3,3,1,2,1],[6,4,5,0,6,6,4,1,5,3,2,2,6,6,1,0] >;

C322SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C3^2:2SD16");
// GroupNames label

G:=SmallGroup(144,118);
// by ID

G=gap.SmallGroup(144,118);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,73,55,218,116,50,964,730,256,299,881]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=b^-1,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^3>;
// generators/relations

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