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G = C3⋊C2≀C4order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C31C2≀C4, C23⋊C41S3, (C2×D4).1D6, (C2×C12).1D4, (C2×C4).1D12, (S3×C23)⋊1C4, C23.6(C4×S3), (C22×C6).8D4, C232D6.1C2, C12.D41C2, C6.D41C4, 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

C1C22×C6 — C3⋊C2≀C4
C1C3C6C2×C6C22×C6C6×D4C232D6 — C3⋊C2≀C4
C3C6C2×C6C22×C6 — C3⋊C2≀C4
C1C2C22C2×D4C23⋊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 [×5], C3, C4 [×3], C22, C22 [×12], S3 [×2], C6, C6 [×3], C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], Dic3, C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C22⋊C4 [×3], M4(2), C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C3×D4, C22×S3 [×4], C22×C6 [×2], 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, C232D6, C3⋊C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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
ρ13222-2-200-12000-111-11003-3-1-33    orthogonal lifted from D12
ρ14222-2-200-12000-111-1100-33-13-3    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 11 6)(2 12 5)(3 8 10)(4 7 9)(13 18 22)(14 19 23)(15 20 24)(16 17 21)
(1 24)(2 10)(3 5)(4 22)(6 15)(7 18)(8 12)(9 13)(11 20)(14 21)(16 23)(17 19)
(1 13)(2 14)(3 16)(4 15)(5 23)(6 22)(7 20)(8 17)(9 24)(10 21)(11 18)(12 19)
(1 4)(6 9)(7 11)(13 15)(18 20)(22 24)
(1 4)(2 3)(5 10)(6 9)(7 11)(8 12)(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)| (1,11,6)(2,12,5)(3,8,10)(4,7,9)(13,18,22)(14,19,23)(15,20,24)(16,17,21), (1,24)(2,10)(3,5)(4,22)(6,15)(7,18)(8,12)(9,13)(11,20)(14,21)(16,23)(17,19), (1,13)(2,14)(3,16)(4,15)(5,23)(6,22)(7,20)(8,17)(9,24)(10,21)(11,18)(12,19), (1,4)(6,9)(7,11)(13,15)(18,20)(22,24), (1,4)(2,3)(5,10)(6,9)(7,11)(8,12)(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( (1,11,6)(2,12,5)(3,8,10)(4,7,9)(13,18,22)(14,19,23)(15,20,24)(16,17,21), (1,24)(2,10)(3,5)(4,22)(6,15)(7,18)(8,12)(9,13)(11,20)(14,21)(16,23)(17,19), (1,13)(2,14)(3,16)(4,15)(5,23)(6,22)(7,20)(8,17)(9,24)(10,21)(11,18)(12,19), (1,4)(6,9)(7,11)(13,15)(18,20)(22,24), (1,4)(2,3)(5,10)(6,9)(7,11)(8,12)(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([(1,11,6),(2,12,5),(3,8,10),(4,7,9),(13,18,22),(14,19,23),(15,20,24),(16,17,21)], [(1,24),(2,10),(3,5),(4,22),(6,15),(7,18),(8,12),(9,13),(11,20),(14,21),(16,23),(17,19)], [(1,13),(2,14),(3,16),(4,15),(5,23),(6,22),(7,20),(8,17),(9,24),(10,21),(11,18),(12,19)], [(1,4),(6,9),(7,11),(13,15),(18,20),(22,24)], [(1,4),(2,3),(5,10),(6,9),(7,11),(8,12),(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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