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G = C23.3D12  order 192 = 26·3

3rd non-split extension by C23 of D12 acting via D12/C3=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.3D12, C3⋊2C2≀C4, (C2×D4).5D6, (S3×C23)⋊2C4, C4.D4⋊5S3, (C2×C12).13D4, C23.8(C4×S3), C23⋊2D6.4C2, C6.D4⋊3C4, (C22×C6).12D4, C6.11(C23⋊C4), C23.7D6⋊7C2, C22.12(D6⋊C4), (C6×D4).170C22, C2.12(C23.6D6), (C2×C4).1(C3⋊D4), (C22×C6).3(C2×C4), (C3×C4.D4)⋊11C2, (C2×C6).5(C22⋊C4), SmallGroup(192,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C6 — C23.3D12
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C6×D4 — C23⋊2D6 — C23.3D12
Lower central
C3 — C6 — C2×C6 — C22×C6 — C23.3D12
Upper central
C1 — C2 — C22 — C2×D4 — C4.D4

Generators and relations for C23.3D12
 G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd11 >

Subgroups: 432 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, C24, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C23⋊C4, C4.D4, C22≀C2, D6⋊C4, C6.D4, C6.D4, C3×M4(2), C2×C3⋊D4, C6×D4, S3×C23, C2≀C4, C23.7D6, C3×C4.D4, C23⋊2D6, C23.3D12
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, C23.3D12

Character table of C23.3D12

 class 12A2B2C2D2E2F34A4B4C4D6A6B6C6D8A8B12A12B24A24B24C24D
 size 11244121224242424248888448888
ρ1111111111111111111111111    trivial
ρ2111111111-1-111111-1-111-1-1-1-1    linear of order 2
ρ311111-1-11111-11111-1-111-1-1-1-1    linear of order 2
ρ411111-1-111-1-1-1111111111111    linear of order 2
ρ51111-1111-1-ii-111-11-ii-1-1i-i-ii    linear of order 4
ρ61111-1111-1i-i-111-11i-i-1-1-iii-i    linear of order 4
ρ71111-1-1-11-1-ii111-11i-i-1-1-iii-i    linear of order 4
ρ81111-1-1-11-1i-i111-11-ii-1-1i-i-ii    linear of order 4
ρ9222-22002-2000222-200-2-20000    orthogonal lifted from D4
ρ102222200-12000-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222-2-2002200022-2-200220000    orthogonal lifted from D4
ρ122222200-12000-1-1-1-1-2-2-1-11111    orthogonal lifted from D6
ρ13222-2200-1-2000-1-1-110011-√3√3-√3√3    orthogonal lifted from D12
ρ14222-2200-1-2000-1-1-110011√3-√3√3-√3    orthogonal lifted from D12
ρ152222-200-1-2000-1-11-1-2i2i11-iii-i    complex lifted from C4×S3
ρ162222-200-1-2000-1-11-12i-2i11i-i-ii    complex lifted from C4×S3
ρ17222-2-200-12000-1-11100-1-1-√-3-√-3√-3√-3    complex lifted from C3⋊D4
ρ18222-2-200-12000-1-11100-1-1√-3√-3-√-3-√-3    complex lifted from C3⋊D4
ρ194-40002-240000-400000000000    orthogonal lifted from C2≀C4
ρ204-4000-2240000-400000000000    orthogonal lifted from C2≀C4
ρ2144-40000400004-40000000000    orthogonal lifted from C23⋊C4
ρ2244-40000-20000-2200002√-3-2√-30000    complex lifted from C23.6D6
ρ2344-40000-20000-220000-2√-32√-30000    complex lifted from C23.6D6
ρ248-800000-40000400000000000    orthogonal faithful

Permutation representations of C23.3D12
►On 24 points - transitive group 24T338
Generators in S24
(1 7)(2 20)(3 21)(4 10)(5 11)(6 24)(8 14)(9 15)(12 18)(13 19)(16 22)(17 23)

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

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

(1 6 7 24)(2 11 20 5)(3 4 21 10)(8 17 14 23)(9 22 15 16)(12 13 18 19)
 
G:=sub<Sym(24)| (1,7)(2,20)(3,21)(4,10)(5,11)(6,24)(8,14)(9,15)(12,18)(13,19)(16,22)(17,23), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,24)(2,11,20,5)(3,4,21,10)(8,17,14,23)(9,22,15,16)(12,13,18,19)>;
 
G:=Group( (1,7)(2,20)(3,21)(4,10)(5,11)(6,24)(8,14)(9,15)(12,18)(13,19)(16,22)(17,23), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,24)(2,11,20,5)(3,4,21,10)(8,17,14,23)(9,22,15,16)(12,13,18,19) );
 
G=PermutationGroup([[(1,7),(2,20),(3,21),(4,10),(5,11),(6,24),(8,14),(9,15),(12,18),(13,19),(16,22),(17,23)], [(2,14),(4,16),(6,18),(8,20),(10,22),(12,24)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,7,24),(2,11,20,5),(3,4,21,10),(8,17,14,23),(9,22,15,16),(12,13,18,19)]])
 
G:=TransitiveGroup(24,338);
 

►On 24 points - transitive group 24T342
Generators in S24
(2 14)(3 15)(6 18)(7 19)(10 22)(11 23)

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

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

(1 24)(2 11 14 23)(3 22 15 10)(4 21)(5 20)(6 7 18 19)(8 17)(9 16)(12 13)
 
G:=sub<Sym(24)| (2,14)(3,15)(6,18)(7,19)(10,22)(11,23), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,11,14,23)(3,22,15,10)(4,21)(5,20)(6,7,18,19)(8,17)(9,16)(12,13)>;
 
G:=Group( (2,14)(3,15)(6,18)(7,19)(10,22)(11,23), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,11,14,23)(3,22,15,10)(4,21)(5,20)(6,7,18,19)(8,17)(9,16)(12,13) );
 
G=PermutationGroup([[(2,14),(3,15),(6,18),(7,19),(10,22),(11,23)], [(2,14),(4,16),(6,18),(8,20),(10,22),(12,24)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,24),(2,11,14,23),(3,22,15,10),(4,21),(5,20),(6,7,18,19),(8,17),(9,16),(12,13)]])
 
G:=TransitiveGroup(24,342);
 

Matrix representation of C23.3D12 ►in GL6(𝔽73)

7200000
0720000
001000
0007200
000010
007207272
,
100000
010000
001000
000100
0000720
007272072
,
100000
010000
0072000
0007200
0000720
0000072
,
30430000
30600000
000010
0072727271
0007200
000101
,
30430000
13430000
0000720
001112
0072000
00072072

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,72,0,0,0,1,0,72,0,0,0,0,72,0,0,0,0,0,0,72],[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],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,0,72,0,0,0,0,0,72,72,1,0,0,1,72,0,0,0,0,0,71,0,1],[30,13,0,0,0,0,43,43,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0,72,0,0,72,1,0,0,0,0,0,2,0,72] >;
 

C23.3D12 in GAP, Magma, Sage, TeX 

C_2^3._3D_{12}
 
% in TeX
 
G:=Group("C2^3.3D12");
 
// GroupNames label
 
G:=SmallGroup(192,34);
 
// by ID
 
G=gap.SmallGroup(192,34);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,346,297,851,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^11>;
 
// generators/relations
 

Export

Character table of C23.3D12 in TeX

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