Copied to
clipboard

G = C3⋊S3⋊2SD16  order 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⋊S3⋊2SD16, (C3×C12).14D4, C32⋊2(C2×SD16), D6⋊D6.6C2, Dic3.D6⋊9C2, C32⋊2C8⋊5C22, C32⋊2Q8⋊1C22, C32⋊2SD16⋊4C2, C3⋊Dic3.5C23, D6⋊S3.1C22, C3⋊S3⋊3C8⋊3C2, (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

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⋊S3⋊2SD16
 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, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×D4, S3×Q8, C32⋊2C8, C6.D6, D6⋊S3, C32⋊2Q8, C3×Dic6, C3×D12, C4×C3⋊S3, C2×S32, C32⋊2SD16, C3⋊S3⋊3C8, Dic3.D6, D6⋊D6, C3⋊S3⋊2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C2×SD16, S3≀C2, C2×S3≀C2, C3⋊S3⋊2SD16

Character table of C3⋊S3⋊2SD16

 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⋊S3⋊2SD16
►On 24 points - transitive group 24T660
Generators in S24
(1 21 9)(3 11 23)(5 17 13)(7 15 19)

(2 22 10)(4 12 24)(6 18 14)(8 16 20)

(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

(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)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)
 
G:=sub<Sym(24)| (1,21,9)(3,11,23)(5,17,13)(7,15,19), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)>;
 
G:=Group( (1,21,9)(3,11,23)(5,17,13)(7,15,19), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24) );
 
G=PermutationGroup([[(1,21,9),(3,11,23),(5,17,13),(7,15,19)], [(2,22,10),(4,12,24),(6,18,14),(8,16,20)], [(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(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),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24)]])
 
G:=TransitiveGroup(24,660);
 

Matrix representation of C3⋊S3⋊2SD16 ►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⋊S3⋊2SD16 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⋊S3⋊2SD16 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁