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G = C23.2D12  order 192 = 26·3

2nd non-split extension by C23 of D12 acting via D12/C3=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.2D12, (C2×D12)⋊3C4, C23⋊C4⋊2S3, (C2×D4).4D6, (C4×Dic3)⋊2C4, C3⋊1(C42⋊C4), C12⋊3D4.1C2, (C6×D4).4C22, (C22×C6).11D4, C6.10(C23⋊C4), C23.7D6⋊1C2, C23.4(C3⋊D4), C22.11(D6⋊C4), C2.11(C23.6D6), (C2×C4).2(C4×S3), (C3×C23⋊C4)⋊2C2, (C2×C12).2(C2×C4), (C2×C6).4(C22⋊C4), SmallGroup(192,33)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C23.2D12
 G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >

Subgroups: 384 in 86 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×D4, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C23⋊C4, C23⋊C4, C4⋊1D4, C4×Dic3, C6.D4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, C6×D4, C42⋊C4, C23.7D6, C3×C23⋊C4, C12⋊3D4, C23.2D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C42⋊C4, C23.6D6, C23.2D12

Character table of C23.2D12

 class 12A2B2C2D2E34A4B4C4D4E4F4G6A6B6C6D6E12A12B12C12D12E
 size 11244242488121224242444888888
ρ1111111111111111111111111    trivial
ρ211111111-1-111-1-111111-1-1-1-11    linear of order 2
ρ311111-111-1-1-1-11111111-1-1-1-11    linear of order 2
ρ411111-11111-1-1-1-11111111111    linear of order 2
ρ5111-1-1-111i-i11-ii1-1-11-1-iii-i1    linear of order 4
ρ6111-1-1-111-ii11i-i1-1-11-1i-i-ii1    linear of order 4
ρ7111-1-1111-ii-1-1-ii1-1-11-1i-i-ii1    linear of order 4
ρ8111-1-1111i-i-1-1i-i1-1-11-1-iii-i1    linear of order 4
ρ92222-202-20000002-2-2220000-2    orthogonal lifted from D4
ρ10222-2202-20000002222-20000-2    orthogonal lifted from D4
ρ11222220-12-2-20000-1-1-1-1-11111-1    orthogonal lifted from D6
ρ12222220-12220000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132222-20-1-2000000-111-1-1√3-√3√3-√31    orthogonal lifted from D12
ρ142222-20-1-2000000-111-1-1-√3√3-√3√31    orthogonal lifted from D12
ρ15222-2-20-122i-2i0000-111-11i-i-ii-1    complex lifted from C4×S3
ρ16222-2-20-12-2i2i0000-111-11-iii-i-1    complex lifted from C4×S3
ρ17222-220-1-2000000-1-1-1-11√-3√-3-√-3-√-31    complex lifted from C3⋊D4
ρ18222-220-1-2000000-1-1-1-11-√-3-√-3√-3√-31    complex lifted from C3⋊D4
ρ1944-400040000000400-4000000    orthogonal lifted from C23⋊C4
ρ204-4000040002-200-4000000000    orthogonal lifted from C42⋊C4
ρ214-400004000-2200-4000000000    orthogonal lifted from C42⋊C4
ρ2244-4000-20000000-22√-3-2√-32000000    complex lifted from C23.6D6
ρ2344-4000-20000000-2-2√-32√-32000000    complex lifted from C23.6D6
ρ248-80000-400000004000000000    orthogonal faithful

Permutation representations of C23.2D12
►On 24 points - transitive group 24T344
Generators in S24
(2 19)(3 20)(6 23)(7 24)(10 15)(11 16)

(1 18)(3 20)(5 22)(7 24)(9 14)(11 16)

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

(1 12)(2 16 19 11)(3 10 20 15)(4 9)(5 8)(6 24 23 7)(13 22)(14 21)(17 18)
 
G:=sub<Sym(24)| (2,19)(3,20)(6,23)(7,24)(10,15)(11,16), (1,18)(3,20)(5,22)(7,24)(9,14)(11,16), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(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), (1,12)(2,16,19,11)(3,10,20,15)(4,9)(5,8)(6,24,23,7)(13,22)(14,21)(17,18)>;
 
G:=Group( (2,19)(3,20)(6,23)(7,24)(10,15)(11,16), (1,18)(3,20)(5,22)(7,24)(9,14)(11,16), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(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), (1,12)(2,16,19,11)(3,10,20,15)(4,9)(5,8)(6,24,23,7)(13,22)(14,21)(17,18) );
 
G=PermutationGroup([[(2,19),(3,20),(6,23),(7,24),(10,15),(11,16)], [(1,18),(3,20),(5,22),(7,24),(9,14),(11,16)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,13),(9,14),(10,15),(11,16),(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)], [(1,12),(2,16,19,11),(3,10,20,15),(4,9),(5,8),(6,24,23,7),(13,22),(14,21),(17,18)]])
 
G:=TransitiveGroup(24,344);
 

►On 24 points - transitive group 24T346
Generators in S24
(1 17)(2 24)(3 13)(4 20)(5 21)(6 16)(7 15)(8 22)(9 23)(10 18)(11 19)(12 14)

(1 9)(3 11)(5 7)(13 19)(15 21)(17 23)

(1 9)(2 10)(3 11)(4 12)(5 7)(6 8)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 17 16)(2 21 24 5)(3 4 13 20)(7 10 15 18)(8 23 22 9)(11 12 19 14)
 
G:=sub<Sym(24)| (1,17)(2,24)(3,13)(4,20)(5,21)(6,16)(7,15)(8,22)(9,23)(10,18)(11,19)(12,14), (1,9)(3,11)(5,7)(13,19)(15,21)(17,23), (1,9)(2,10)(3,11)(4,12)(5,7)(6,8)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,17,16)(2,21,24,5)(3,4,13,20)(7,10,15,18)(8,23,22,9)(11,12,19,14)>;
 
G:=Group( (1,17)(2,24)(3,13)(4,20)(5,21)(6,16)(7,15)(8,22)(9,23)(10,18)(11,19)(12,14), (1,9)(3,11)(5,7)(13,19)(15,21)(17,23), (1,9)(2,10)(3,11)(4,12)(5,7)(6,8)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,17,16)(2,21,24,5)(3,4,13,20)(7,10,15,18)(8,23,22,9)(11,12,19,14) );
 
G=PermutationGroup([[(1,17),(2,24),(3,13),(4,20),(5,21),(6,16),(7,15),(8,22),(9,23),(10,18),(11,19),(12,14)], [(1,9),(3,11),(5,7),(13,19),(15,21),(17,23)], [(1,9),(2,10),(3,11),(4,12),(5,7),(6,8),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,17,16),(2,21,24,5),(3,4,13,20),(7,10,15,18),(8,23,22,9),(11,12,19,14)]])
 
G:=TransitiveGroup(24,346);
 

Matrix representation of C23.2D12 ►in GL6(ℤ)

100000
010000
000-100
00-1000
000001
000010
,
100000
010000
001000
000100
0000-10
00000-1
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
0-10000
1-10000
000010
000001
001000
000-100
,
1-10000
0-10000
0000-10
000001
000100
00-1000

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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,-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,-1,0,0,0,0,0,0,-1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,0,0] >;
 

C23.2D12 in GAP, Magma, Sage, TeX 

C_2^3._2D_{12}
 
% in TeX
 
G:=Group("C2^3.2D12");
 
// GroupNames label
 
G:=SmallGroup(192,33);
 
// by ID
 
G=gap.SmallGroup(192,33);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,1123,794,297,136,851,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=a,d*a*d^-1=a*b=b*a,a*c=c*a,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*d^-1>;
 
// generators/relations
 

Export

Character table of C23.2D12 in TeX

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