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G = C3⋊S3⋊2SD16order 288 = 25·32

The semidirect product of C3⋊S3 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3⋊S3⋊2SD16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊2SD16 — C3⋊S3⋊2SD16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3⋊S3⋊2SD16
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S32SD16
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, cac=a-1, dad-1=cbc=ebe=b-1, ae=ea, dbd-1=a, cd=dc, ce=ec, ede=d3 >

Subgroups: 656 in 114 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×3], C22 [×5], S3 [×6], C6 [×4], C8 [×2], C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3 [×4], C12 [×4], D6 [×8], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×3], C4×S3 [×4], D12, C3⋊D4 [×2], C3×D4, C3×Q8, C22×S3 [×2], C2×SD16, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×D4, S3×Q8, C322C8 [×2], C6.D6, D6⋊S3 [×2], C322Q8 [×2], C3×Dic6, C3×D12, C4×C3⋊S3, C2×S32, C322SD16 [×4], C3⋊S33C8, Dic3.D6, D6⋊D6, C3⋊S32SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C2×SD16, S3≀C2, C2×S3≀C2, C3⋊S32SD16

Character table of C3⋊S32SD16

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 9 9 12 12 4 4 2 12 12 18 4 4 24 24 18 18 18 18 8 8 24 24 ρ1 1 1 1 1 1 1 1 1 1 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 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 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 1 -1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ8 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 0 0 2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 2 2 2 0 0 -2 2 2 0 0 0 0 0 0 2 2 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ12 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 -2 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ13 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 -2 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ14 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ15 4 4 0 0 -2 -2 1 -2 4 0 0 0 1 -2 1 1 0 0 0 0 1 -2 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 0 0 -2 2 1 -2 -4 0 0 0 1 -2 -1 1 0 0 0 0 -1 2 0 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 2 -2 1 -2 -4 0 0 0 1 -2 1 -1 0 0 0 0 -1 2 0 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 0 0 0 -2 1 -4 -2 2 0 -2 1 0 0 0 0 0 0 2 -1 -1 1 orthogonal lifted from C2×S3≀C2 ρ19 4 4 0 0 2 2 1 -2 4 0 0 0 1 -2 -1 -1 0 0 0 0 1 -2 0 0 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 0 0 -2 1 4 -2 -2 0 -2 1 0 0 0 0 0 0 -2 1 1 1 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 0 0 -2 1 4 2 2 0 -2 1 0 0 0 0 0 0 -2 1 -1 -1 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 0 0 -2 1 -4 2 -2 0 -2 1 0 0 0 0 0 0 2 -1 1 -1 orthogonal lifted from C2×S3≀C2 ρ23 8 -8 0 0 0 0 2 -4 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 8 -8 0 0 0 0 -4 2 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C3⋊S32SD16
On 24 points - transitive group 24T660
Generators in S24
(1 21 11)(3 13 23)(5 17 15)(7 9 19)
(2 22 12)(4 14 24)(6 18 16)(8 10 20)
(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 13)(10 16)(12 14)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,660);

Matrix representation of C3⋊S32SD16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 0 18 0 0 0 0 69 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 1 1 0 0
,
 1 0 0 0 0 0 25 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 1 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,69,0,0,0,0,18,12,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,25,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;

C3⋊S32SD16 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C3:S3:2SD16");
// GroupNames label

G:=SmallGroup(288,875);
// by ID

G=gap.SmallGroup(288,875);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,219,100,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

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

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