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G = C23⋊D12  order 192 = 26·3

The semidirect product of C23 and D12 acting via D12/C3=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊D12, (C2×C4)⋊D12, C3⋊1C2≀C22, (C2×C12)⋊1D4, C23⋊C4⋊3S3, C22⋊C4⋊1D6, (C22×C6)⋊2D4, C23⋊2D6⋊1C2, D4⋊6D6⋊1C2, D6⋊D4⋊1C2, (C2×Dic3)⋊1D4, (C2×D4).11D6, (C22×S3)⋊1D4, C6.13C22≀C2, (C6×D4).8C22, C22.24(S3×D4), C22.8(C2×D12), (S3×C23)⋊1C22, C23.6D6⋊1C2, (C22×C6).2C23, C2.16(D6⋊D4), C6.D4⋊1C22, C23.12(C22×S3), (C3×C23⋊C4)⋊4C2, (C2×C6).17(C2×D4), (C3×C22⋊C4)⋊1C22, (C2×C3⋊D4).2C22, SmallGroup(192,300)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C6 — C23⋊D12
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — D4⋊6D6 — C23⋊D12
Lower central
C3 — C6 — C22×C6 — C23⋊D12
Upper central
C1 — C2 — C23 — C23⋊C4

Generators and relations for C23⋊D12
 G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 800 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C23⋊C4, C23⋊C4, C22≀C2, 2+ 1+4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C2≀C22, C23.6D6, C3×C23⋊C4, D6⋊D4, C23⋊2D6, D4⋊6D6, C23⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, C2≀C22, D6⋊D4, C23⋊D12

Character table of C23⋊D12

 class 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F6A6B6C6D6E12A12B12C12D12E
 size 1122241212121224881212242444888888
ρ1111111111111111111111111111    trivial
ρ2111111-1-1-1-11111-1-1-11111111111    linear of order 2
ρ3111111-1-11111-1-1-1-1111111-1-1-11-1    linear of order 2
ρ411111111-1-111-1-111-111111-1-1-11-1    linear of order 2
ρ511111-11-1111-1-111-1-11111-1-11-1-11    linear of order 2
ρ611111-1-11-1-11-1-11-1111111-1-11-1-11    linear of order 2
ρ711111-1-11111-11-1-11-11111-11-11-1-1    linear of order 2
ρ811111-11-1-1-11-11-11-111111-11-11-1-1    linear of order 2
ρ922-22-20020020000-2022-2-2000000    orthogonal lifted from D4
ρ1022-2-220-200020002002-2-22000000    orthogonal lifted from D4
ρ1122222-20000-1-2-22000-1-1-1-111-111-1    orthogonal lifted from D6
ρ1222222-20000-1-22-2000-1-1-1-11-11-111    orthogonal lifted from D6
ρ1322-2-22020002000-2002-2-22000000    orthogonal lifted from D4
ρ1422-22-200-200200002022-2-2000000    orthogonal lifted from D4
ρ152222220000-12-2-2000-1-1-1-1-1111-11    orthogonal lifted from D6
ρ162222220000-1222000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ17222-2-2200002-2000002-22-22000-20    orthogonal lifted from D4
ρ18222-2-2-2000022000002-22-2-200020    orthogonal lifted from D4
ρ19222-2-220000-1-200000-11-11-1√3√3-√31-√3    orthogonal lifted from D12
ρ20222-2-2-20000-1200000-11-111√3-√3-√3-1√3    orthogonal lifted from D12
ρ21222-2-2-20000-1200000-11-111-√3√3√3-1-√3    orthogonal lifted from D12
ρ22222-2-220000-1-200000-11-11-1-√3-√3√31√3    orthogonal lifted from D12
ρ234-40000002-24000000-4000000000    orthogonal lifted from C2≀C22
ρ2444-4-4400000-2000000-222-2000000    orthogonal lifted from S3×D4
ρ2544-44-400000-2000000-2-222000000    orthogonal lifted from S3×D4
ρ264-4000000-224000000-4000000000    orthogonal lifted from C2≀C22
ρ278-800000000-40000004000000000    orthogonal faithful

Permutation representations of C23⋊D12
►On 24 points - transitive group 24T336
Generators in S24
(1 16)(2 17)(3 24)(4 13)(5 20)(6 21)(7 19)(8 14)(9 15)(10 22)(11 23)(12 18)

(1 10)(3 12)(5 8)(14 20)(16 22)(18 24)

(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(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 3)(4 6)(7 9)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)
 
G:=sub<Sym(24)| (1,16)(2,17)(3,24)(4,13)(5,20)(6,21)(7,19)(8,14)(9,15)(10,22)(11,23)(12,18), (1,10)(3,12)(5,8)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(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), (1,3)(4,6)(7,9)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21)>;
 
G:=Group( (1,16)(2,17)(3,24)(4,13)(5,20)(6,21)(7,19)(8,14)(9,15)(10,22)(11,23)(12,18), (1,10)(3,12)(5,8)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(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), (1,3)(4,6)(7,9)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21) );
 
G=PermutationGroup([[(1,16),(2,17),(3,24),(4,13),(5,20),(6,21),(7,19),(8,14),(9,15),(10,22),(11,23),(12,18)], [(1,10),(3,12),(5,8),(14,20),(16,22),(18,24)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(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)], [(1,3),(4,6),(7,9),(10,12),(13,15),(16,24),(17,23),(18,22),(19,21)]])
 
G:=TransitiveGroup(24,336);
 

►On 24 points - transitive group 24T345
Generators in S24
(2 20)(3 21)(6 24)(7 13)(10 16)(11 17)

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

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

(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)
 
G:=sub<Sym(24)| (2,20)(3,21)(6,24)(7,13)(10,16)(11,17), (2,20)(4,22)(6,24)(8,14)(10,16)(12,18), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)>;
 
G:=Group( (2,20)(3,21)(6,24)(7,13)(10,16)(11,17), (2,20)(4,22)(6,24)(8,14)(10,16)(12,18), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21) );
 
G=PermutationGroup([[(2,20),(3,21),(6,24),(7,13),(10,16),(11,17)], [(2,20),(4,22),(6,24),(8,14),(10,16),(12,18)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21)]])
 
G:=TransitiveGroup(24,345);
 

►On 24 points - transitive group 24T367
Generators in S24
(1 19)(2 23)(3 15)(4 13)(5 17)(6 21)(7 16)(8 14)(9 24)(10 22)(11 20)(12 18)

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

(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(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)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)
 
G:=sub<Sym(24)| (1,19)(2,23)(3,15)(4,13)(5,17)(6,21)(7,16)(8,14)(9,24)(10,22)(11,20)(12,18), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(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)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;
 
G:=Group( (1,19)(2,23)(3,15)(4,13)(5,17)(6,21)(7,16)(8,14)(9,24)(10,22)(11,20)(12,18), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(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)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );
 
G=PermutationGroup([[(1,19),(2,23),(3,15),(4,13),(5,17),(6,21),(7,16),(8,14),(9,24),(10,22),(11,20),(12,18)], [(1,7),(2,11),(3,9),(4,10),(5,8),(6,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(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),(2,5),(3,4),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)]])
 
G:=TransitiveGroup(24,367);
 

Matrix representation of C23⋊D12 ►in GL6(𝔽13)

100000
010000
000010
000001
001000
000100
,
100000
010000
000100
001000
000001
000010
,
100000
010000
0012000
0001200
0000120
0000012
,
870000
1290000
001000
0001200
0000012
000010
,
470000
990000
001000
000100
0000012
0000120

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,12,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[4,9,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
 

C23⋊D12 in GAP, Magma, Sage, TeX 

C_2^3\rtimes D_{12}
 
% in TeX
 
G:=Group("C2^3:D12");
 
// GroupNames label
 
G:=SmallGroup(192,300);
 
// by ID
 
G=gap.SmallGroup(192,300);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,438,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C23⋊D12 in TeX

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