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G = C3⋊C2≀C4  order 192 = 26·3

The semidirect product of C3 and C2≀C4 acting via C2≀C4/C23⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊1C2≀C4, C23⋊C4⋊1S3, (C2×D4).1D6, (C2×C12).1D4, (C2×C4).1D12, (S3×C23)⋊1C4, C23.6(C4×S3), (C22×C6).8D4, C23⋊2D6.1C2, C12.D4⋊1C2, C6.D4⋊1C4, C6.7(C23⋊C4), (C6×D4).1C22, C22.8(D6⋊C4), C23.1(C3⋊D4), C2.8(C23.6D6), (C3×C23⋊C4)⋊1C2, (C22×C6).1(C2×C4), (C2×C6).1(C22⋊C4), SmallGroup(192,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C6 — C3⋊C2≀C4
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23⋊2D6 — C3⋊C2≀C4
Lower central
C3 — C6 — C2×C6 — C22×C6 — C3⋊C2≀C4
Upper central
C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for C3⋊C2≀C4
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 416 in 94 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, M4(2), C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C23⋊C4, C4.D4, C22≀C2, C4.Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C6×D4, S3×C23, C2≀C4, C12.D4, C3×C23⋊C4, C23⋊2D6, C3⋊C2≀C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C2≀C4, C23.6D6, C3⋊C2≀C4

Character table of C3⋊C2≀C4

 class 12A2B2C2D2E2F34A4B4C4D6A6B6C6D6E8A8B12A12B12C12D12E
 size 11244121224882424448242488888
ρ1111111111111111111111111    trivial
ρ211111-1-111-1-1-11111111-1-11-1-1    linear of order 2
ρ3111111111-1-1111111-1-1-1-11-1-1    linear of order 2
ρ411111-1-11111-111111-1-111111    linear of order 2
ρ5111-11-1-11-1-ii11-1-111i-iii-1-i-i    linear of order 4
ρ6111-11111-1i-i-11-1-111i-i-i-i-1ii    linear of order 4
ρ7111-11-1-11-1i-i11-1-111-ii-i-i-1ii    linear of order 4
ρ8111-11111-1-ii-11-1-111-iiii-1-i-i    linear of order 4
ρ92222200-12220-1-1-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ102222-2002-20002222-20000-200    orthogonal lifted from D4
ρ112222200-12-2-20-1-1-1-1-10011-111    orthogonal lifted from D6
ρ12222-2-200220002-2-22-20000200    orthogonal lifted from D4
ρ13222-2-200-12000-111-1100√3-√3-1-√3√3    orthogonal lifted from D12
ρ14222-2-200-12000-111-1100-√3√3-1√3-√3    orthogonal lifted from D12
ρ15222-2200-1-2-2i2i0-111-1-100-i-i1ii    complex lifted from C4×S3
ρ16222-2200-1-22i-2i0-111-1-100ii1-i-i    complex lifted from C4×S3
ρ172222-200-1-2000-1-1-1-1100-√-3√-31-√-3√-3    complex lifted from C3⋊D4
ρ182222-200-1-2000-1-1-1-1100√-3-√-31√-3-√-3    complex lifted from C3⋊D4
ρ1944-4000040000400-400000000    orthogonal lifted from C23⋊C4
ρ204-40002-240000-400000000000    orthogonal lifted from C2≀C4
ρ214-4000-2240000-400000000000    orthogonal lifted from C2≀C4
ρ2244-40000-20000-2-2√-32√-3200000000    complex lifted from C23.6D6
ρ2344-40000-20000-22√-3-2√-3200000000    complex lifted from C23.6D6
ρ248-800000-40000400000000000    orthogonal faithful

Permutation representations of C3⋊C2≀C4
►On 24 points - transitive group 24T340
Generators in S24
(1 9 6)(2 10 7)(3 11 8)(4 12 5)(13 17 22)(14 18 23)(15 19 24)(16 20 21)

(2 15)(5 12)(6 9)(7 19)(8 11)(10 24)(17 22)(18 23)(20 21)

(2 15)(3 16)(7 24)(8 21)(10 19)(11 20)

(2 15)(4 13)(5 22)(7 24)(10 19)(12 17)

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

(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)| (1,9,6)(2,10,7)(3,11,8)(4,12,5)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (2,15)(5,12)(6,9)(7,19)(8,11)(10,24)(17,22)(18,23)(20,21), (2,15)(3,16)(7,24)(8,21)(10,19)(11,20), (2,15)(4,13)(5,22)(7,24)(10,19)(12,17), (1,14)(2,15)(3,16)(4,13)(5,22)(6,23)(7,24)(8,21)(9,18)(10,19)(11,20)(12,17), (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( (1,9,6)(2,10,7)(3,11,8)(4,12,5)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (2,15)(5,12)(6,9)(7,19)(8,11)(10,24)(17,22)(18,23)(20,21), (2,15)(3,16)(7,24)(8,21)(10,19)(11,20), (2,15)(4,13)(5,22)(7,24)(10,19)(12,17), (1,14)(2,15)(3,16)(4,13)(5,22)(6,23)(7,24)(8,21)(9,18)(10,19)(11,20)(12,17), (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([[(1,9,6),(2,10,7),(3,11,8),(4,12,5),(13,17,22),(14,18,23),(15,19,24),(16,20,21)], [(2,15),(5,12),(6,9),(7,19),(8,11),(10,24),(17,22),(18,23),(20,21)], [(2,15),(3,16),(7,24),(8,21),(10,19),(11,20)], [(2,15),(4,13),(5,22),(7,24),(10,19),(12,17)], [(1,14),(2,15),(3,16),(4,13),(5,22),(6,23),(7,24),(8,21),(9,18),(10,19),(11,20),(12,17)], [(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(24,340);
 

►On 24 points - transitive group 24T348
Generators in S24
(1 3 6)(2 4 5)(7 9 11)(8 10 12)(13 23 18)(14 24 19)(15 21 20)(16 22 17)

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

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

(2 9)(4 11)(5 7)(13 15)(18 20)(21 23)

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

(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)| (1,3,6)(2,4,5)(7,9,11)(8,10,12)(13,23,18)(14,24,19)(15,21,20)(16,22,17), (1,10)(2,13)(3,8)(4,18)(5,23)(6,12)(7,21)(9,15)(11,20)(14,16)(17,24)(19,22), (1,16)(2,15)(3,22)(4,21)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,23)(12,24), (2,9)(4,11)(5,7)(13,15)(18,20)(21,23), (1,10)(2,9)(3,12)(4,11)(5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (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( (1,3,6)(2,4,5)(7,9,11)(8,10,12)(13,23,18)(14,24,19)(15,21,20)(16,22,17), (1,10)(2,13)(3,8)(4,18)(5,23)(6,12)(7,21)(9,15)(11,20)(14,16)(17,24)(19,22), (1,16)(2,15)(3,22)(4,21)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,23)(12,24), (2,9)(4,11)(5,7)(13,15)(18,20)(21,23), (1,10)(2,9)(3,12)(4,11)(5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (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([[(1,3,6),(2,4,5),(7,9,11),(8,10,12),(13,23,18),(14,24,19),(15,21,20),(16,22,17)], [(1,10),(2,13),(3,8),(4,18),(5,23),(6,12),(7,21),(9,15),(11,20),(14,16),(17,24),(19,22)], [(1,16),(2,15),(3,22),(4,21),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,23),(12,24)], [(2,9),(4,11),(5,7),(13,15),(18,20),(21,23)], [(1,10),(2,9),(3,12),(4,11),(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(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(24,348);
 

Matrix representation of C3⋊C2≀C4 ►in GL6(𝔽73)

010000
72720000
001000
000100
000010
000001
,
100000
72720000
001000
000100
0000148
0000072
,
100000
010000
0014800
0007200
0000148
0000072
,
100000
010000
001000
000100
0000720
0000072
,
100000
010000
0072000
0007200
0000720
0000072
,
2700000
0270000
000010
000001
001000
0037200

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,48,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,1,0,0,0,0,0,48,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C3⋊C2≀C4 in GAP, Magma, Sage, TeX 

C_3\rtimes C_2\wr C_4
 
% in TeX
 
G:=Group("C3:C2wrC4");
 
// GroupNames label
 
G:=SmallGroup(192,30);
 
// by ID
 
G=gap.SmallGroup(192,30);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,346,297,851,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
 
// generators/relations
 

Export

Character table of C3⋊C2≀C4 in TeX

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