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G = C3⋊S32SD16order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C4.7S3≀C2, C3⋊S32SD16, (C3×C12).14D4, C322(C2×SD16), D6⋊D6.6C2, Dic3.D69C2, C322C85C22, C322Q81C22, C322SD164C2, C3⋊Dic3.5C23, D6⋊S3.1C22, C3⋊S33C83C2, (C3×C6).8(C2×D4), C2.11(C2×S3≀C2), (C2×C3⋊S3).32D4, (C4×C3⋊S3).33C22, SmallGroup(288,875)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3⋊S32SD16
C1C32C3×C6C3⋊Dic3D6⋊S3C322SD16 — C3⋊S32SD16
C32C3×C6C3⋊Dic3 — C3⋊S32SD16
C1C2C4

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 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D8A8B8C8D12A12B12C12D
 size 1199121244212121844242418181818882424
ρ1111111111111111111111111    trivial
ρ2111111111-1-111111-1-1-1-111-1-1    linear of order 2
ρ311-1-11-111-11-1111-111-1-11-1-1-11    linear of order 2
ρ411-1-11-111-1-11111-11-111-1-1-11-1    linear of order 2
ρ51111-1-111111111-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1111-1-1111-1-1111111-1-1    linear of order 2
ρ711-1-1-1111-11-11111-1-111-1-1-1-11    linear of order 2
ρ811-1-1-1111-1-111111-11-1-11-1-11-1    linear of order 2
ρ922220022-200-222000000-2-200    orthogonal lifted from D4
ρ1022-2-20022200-2220000002200    orthogonal lifted from D4
ρ112-2-2200220000-2-200--2--2-2-20000    complex lifted from SD16
ρ122-22-200220000-2-200--2-2--2-20000    complex lifted from SD16
ρ132-22-200220000-2-200-2--2-2--20000    complex lifted from SD16
ρ142-2-2200220000-2-200-2-2--2--20000    complex lifted from SD16
ρ154400-2-21-240001-21100001-200    orthogonal lifted from S3≀C2
ρ164400-221-2-40001-2-110000-1200    orthogonal lifted from C2×S3≀C2
ρ1744002-21-2-40001-21-10000-1200    orthogonal lifted from C2×S3≀C2
ρ18440000-21-4-220-210000002-1-11    orthogonal lifted from C2×S3≀C2
ρ194400221-240001-2-1-100001-200    orthogonal lifted from S3≀C2
ρ20440000-214-2-20-21000000-2111    orthogonal lifted from S3≀C2
ρ21440000-214220-21000000-21-1-1    orthogonal lifted from S3≀C2
ρ22440000-21-42-20-210000002-11-1    orthogonal lifted from C2×S3≀C2
ρ238-800002-40000-240000000000    orthogonal faithful
ρ248-80000-4200004-20000000000    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)

100000
010000
000100
00727200
000010
000001
,
100000
010000
001000
000100
00007272
000010
,
100000
010000
001000
00727200
000010
00007272
,
0180000
69120000
000010
000001
0072000
001100
,
100000
25720000
001000
000100
0000720
000011

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

Export

Character table of C3⋊S32SD16 in TeX

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