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G = C3⋊S3⋊3C8  order 144 = 24·32

2nd semidirect product of C3⋊S3 and C8 acting via C8/C4=C2

metabelian, soluble, monomial, A-group

Aliases: C3⋊S3⋊3C8, C32⋊2(C2×C8), (C3×C12).2C4, C32⋊2C8⋊5C2, C4.3(C32⋊C4), C3⋊Dic3.7C22, (C4×C3⋊S3).6C2, (C2×C3⋊S3).5C4, (C3×C6).1(C2×C4), C2.1(C2×C32⋊C4), SmallGroup(144,130)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊S3⋊3C8
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C3⋊S3⋊3C8
Lower central
C32 — C3⋊S3⋊3C8
Upper central
C1 — C4

Generators and relations for C3⋊S3⋊3C8
 G = < a,b,c,d | a3=b3=c2=d8=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a-1b-1, cd=dc >

C1
C2
9C2
9C2
2C3
2C3
C4
9C4
9C22
2C6
2C6
6S3
6S3
6S3
6S3
C32
9C8
9C8
9C2×C4
2C12
2C12
6D6
6Dic3
6Dic3
6D6
C3⋊S3
C3×C6
C3⋊S3
9C2×C8
6C4×S3
6C4×S3
C3⋊Dic3
C3×C12
C2×C3⋊S3
C32⋊2C8
C4×C3⋊S3
C32⋊2C8
C3⋊S3⋊3C8

Character table of C3⋊S3⋊3C8

 class 12A2B2C3A3B4A4B4C4D6A6B8A8B8C8D8E8F8G8H12A12B12C12D
 size 119944119944999999994444
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ311-1-111-1-11111-1111-1-1-11-1-1-1-1    linear of order 2
ρ411-1-111-1-111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ511-1-11111-1-111i-iii-i-ii-i1111    linear of order 4
ρ6111111-1-1-1-111ii-i-i-i-iii-1-1-1-1    linear of order 4
ρ711-1-11111-1-111-ii-i-iii-ii1111    linear of order 4
ρ8111111-1-1-1-111-i-iiiii-i-i-1-1-1-1    linear of order 4
ρ91-11-111i-ii-i-1-1ζ8ζ85ζ83ζ87ζ83ζ87ζ85ζ8-i-iii    linear of order 8
ρ101-1-1111i-i-ii-1-1ζ87ζ87ζ8ζ85ζ85ζ8ζ83ζ83-i-iii    linear of order 8
ρ111-1-1111-iii-i-1-1ζ8ζ8ζ87ζ83ζ83ζ87ζ85ζ85ii-i-i    linear of order 8
ρ121-11-111-ii-ii-1-1ζ87ζ83ζ85ζ8ζ85ζ8ζ83ζ87ii-i-i    linear of order 8
ρ131-1-1111i-i-ii-1-1ζ83ζ83ζ85ζ8ζ8ζ85ζ87ζ87-i-iii    linear of order 8
ρ141-1-1111-iii-i-1-1ζ85ζ85ζ83ζ87ζ87ζ83ζ8ζ8ii-i-i    linear of order 8
ρ151-11-111i-ii-i-1-1ζ85ζ8ζ87ζ83ζ87ζ83ζ8ζ85-i-iii    linear of order 8
ρ161-11-111-ii-ii-1-1ζ83ζ87ζ8ζ85ζ8ζ85ζ87ζ83ii-i-i    linear of order 8
ρ174400-214400-2100000000-211-2    orthogonal lifted from C32⋊C4
ρ184400-21-4-400-21000000002-1-12    orthogonal lifted from C2×C32⋊C4
ρ1944001-244001-2000000001-2-21    orthogonal lifted from C32⋊C4
ρ2044001-2-4-4001-200000000-122-1    orthogonal lifted from C2×C32⋊C4
ρ214-4001-2-4i4i00-1200000000i-2i2i-i    complex faithful
ρ224-4001-24i-4i00-1200000000-i2i-2ii    complex faithful
ρ234-400-21-4i4i002-100000000-2ii-i2i    complex faithful
ρ244-400-214i-4i002-1000000002i-ii-2i    complex faithful

Permutation representations of C3⋊S3⋊3C8
►On 24 points - transitive group 24T244
Generators in S24
(1 21 13)(2 22 14)(3 15 23)(4 16 24)(5 17 9)(6 18 10)(7 11 19)(8 12 20)

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

(1 5)(2 6)(3 7)(4 8)(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)
 
G:=sub<Sym(24)| (1,21,13)(2,22,14)(3,15,23)(4,16,24)(5,17,9)(6,18,10)(7,11,19)(8,12,20), (2,14,22)(4,24,16)(6,10,18)(8,20,12), (1,5)(2,6)(3,7)(4,8)(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)>;
 
G:=Group( (1,21,13)(2,22,14)(3,15,23)(4,16,24)(5,17,9)(6,18,10)(7,11,19)(8,12,20), (2,14,22)(4,24,16)(6,10,18)(8,20,12), (1,5)(2,6)(3,7)(4,8)(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) );
 
G=PermutationGroup([[(1,21,13),(2,22,14),(3,15,23),(4,16,24),(5,17,9),(6,18,10),(7,11,19),(8,12,20)], [(2,14,22),(4,24,16),(6,10,18),(8,20,12)], [(1,5),(2,6),(3,7),(4,8),(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)]])
 
G:=TransitiveGroup(24,244);
 

C3⋊S3⋊3C8 is a maximal subgroup of
 S32⋊C8  C4.19S3≀C2  C4.4PSU3(𝔽2)  C4.PSU3(𝔽2)  C4.2PSU3(𝔽2)  C8×C32⋊C4  (C3×C24)⋊C4  C3⋊S3.5D8  C3⋊S3.5Q16  C4.3F9  C4.F9  C32⋊D8⋊5C2  C3⋊S3⋊D8  C3⋊S3⋊2SD16  C3⋊S3⋊Q16  C3⋊S3⋊M4(2)  C62.(C2×C4)  C12⋊S3.C4  C33⋊5(C2×C8)  C33⋊7(C2×C8)
C3⋊S3⋊3C8 is a maximal quotient of
 C3⋊S3⋊3C16  C32⋊3M5(2)  C4×C32⋊2C8  C62.6(C2×C4)  C32⋊5(C4⋊C8)  He3⋊2(C2×C8)  C33⋊5(C2×C8)  C33⋊7(C2×C8)

Matrix representation of C3⋊S3⋊3C8 ►in GL4(𝔽5) generated by

2130
1214
4002
4314
,
0043
3000
0303
0001
,
2423
1340
0243
1241
,
1002
3004
0031
0131
G:=sub<GL(4,GF(5))| [2,1,4,4,1,2,0,3,3,1,0,1,0,4,2,4],[0,3,0,0,0,0,3,0,4,0,0,0,3,0,3,1],[2,1,0,1,4,3,2,2,2,4,4,4,3,0,3,1],[1,3,0,0,0,0,0,1,0,0,3,3,2,4,1,1] >;
 

C3⋊S3⋊3C8 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes_3C_8
 
% in TeX
 
G:=Group("C3:S3:3C8");
 
// GroupNames label
 
G:=SmallGroup(144,130);
 
// by ID
 
G=gap.SmallGroup(144,130);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,55,50,3364,256,4613,881]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^2=d^8=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;
 
// generators/relations
 

Export

Subgroup lattice of C3⋊S3⋊3C8 in TeX
Character table of C3⋊S3⋊3C8 in TeX

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