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G = C2xC3:D12  order 144 = 24·32

Direct product of C2 and C3:D12

direct product, metabelian, supersoluble, monomial

Aliases: C2xC3:D12, D6:5D6, C6:2D12, Dic3:4D6, C62.11C22, (C3xC6):3D4, C3:3(C2xD12), C32:5(C2xD4), C6:1(C3:D4), C22.11S32, (C2xC6).16D6, (C22xS3):3S3, (S3xC6):6C22, (C6xDic3):5C2, (C2xDic3):4S3, (C3xC6).15C23, C6.15(C22xS3), (C3xDic3):4C22, (S3xC2xC6):2C2, C2.15(C2xS32), C3:1(C2xC3:D4), (C2xC3:S3):3C22, (C22xC3:S3):1C2, SmallGroup(144,151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C2xC3:D12
Chief series
C1 — C3 — C32 — C3xC6 — S3xC6 — C3:D12 — C2xC3:D12
Lower central
C32 — C3xC6 — C2xC3:D12
Upper central
C1 — C22

Generators and relations for C2xC3:D12
 G = < a,b,c,d | a2=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 464 in 124 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xS3, C22xC6, C3xDic3, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C2xD12, C2xC3:D4, C3:D12, C6xDic3, S3xC2xC6, C22xC3:S3, C2xC3:D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C3:D4, C22xS3, S32, C2xD12, C2xC3:D4, C3:D12, C2xS32, C2xC3:D12

Character table of C2xC3:D12

 class 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D
 size 11116618182246622222244466666666
ρ1111111111111111111111111111111    trivial
ρ2111111-1-1111-1-11111111111111-1-1-1-1    linear of order 2
ρ31111-1-1-1-111111111111111-1-1-1-11111    linear of order 2
ρ41111-1-111111-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-11-111-1111-11-1-1-1-1111-1-11-1-1111-1-1    linear of order 2
ρ61-1-11-11-111111-1-1-1-1-1111-1-11-1-11-1-111    linear of order 2
ρ71-1-111-1-11111-11-1-1-1-1111-1-1-111-111-1-1    linear of order 2
ρ81-1-111-11-11111-1-1-1-1-1111-1-1-111-1-1-111    linear of order 2
ρ92222-2-2002-1-1002-1-122-1-1-1-111110000    orthogonal lifted from D6
ρ1022220000-12-1-2-2-122-1-12-1-1-100001111    orthogonal lifted from D6
ρ112-2-220000-12-12-21-2-21-12-111000011-1-1    orthogonal lifted from D6
ρ122-2-220000-12-1-221-2-21-12-1110000-1-111    orthogonal lifted from D6
ρ1322220000-12-122-122-1-12-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ14222222002-1-1002-1-122-1-1-1-1-1-1-1-10000    orthogonal lifted from S3
ρ152-2-22-22002-1-100-211-22-1-111-111-10000    orthogonal lifted from D6
ρ162-22-2000022200-2-222-2-2-2-2200000000    orthogonal lifted from D4
ρ1722-2-200002220022-2-2-2-2-22-200000000    orthogonal lifted from D4
ρ182-2-222-2002-1-100-211-22-1-1111-1-110000    orthogonal lifted from D6
ρ1922-2-20000-12-100-12-211-21-110000-√3√3-√3√3    orthogonal lifted from D12
ρ202-22-20000-12-1001-22-11-211-10000-√3√3√3-√3    orthogonal lifted from D12
ρ2122-2-20000-12-100-12-211-21-110000√3-√3√3-√3    orthogonal lifted from D12
ρ222-22-20000-12-1001-22-11-211-10000√3-√3-√3√3    orthogonal lifted from D12
ρ2322-2-200002-1-1002-11-2-211-11-√-3-√-3√-3√-30000    complex lifted from C3:D4
ρ2422-2-200002-1-1002-11-2-211-11√-3√-3-√-3-√-30000    complex lifted from C3:D4
ρ252-22-200002-1-100-21-12-2111-1√-3-√-3√-3-√-30000    complex lifted from C3:D4
ρ262-22-200002-1-100-21-12-2111-1-√-3√-3-√-3√-30000    complex lifted from C3:D4
ρ274-44-40000-2-210022-2-222-1-1100000000    orthogonal lifted from C3:D12
ρ2844440000-2-2100-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ294-4-440000-2-21002222-2-21-1-100000000    orthogonal lifted from C2xS32
ρ3044-4-40000-2-2100-2-22222-11-100000000    orthogonal lifted from C3:D12

Permutation representations of C2xC3:D12
►On 24 points - transitive group 24T230
Generators in S24
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)

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

(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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)
 
G:=sub<Sym(24)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)>;
 
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17) );
 
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,17,21),(14,22,18),(15,19,23),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17)]])
 
G:=TransitiveGroup(24,230);
 

C2xC3:D12 is a maximal subgroup of
 C62.20C23  Dic3.D12  C62.23C23  C62.24C23  Dic3:4D12  C62.51C23  Dic3:D12  D6.D12  C62.74C23  D6:D12  C62.77C23  C12:7D12  Dic3:3D12  C12:D12  C62.82C23  C12:2D12  D6:4D12  D6:5D12  C62.100C23  C62.113C23  C62:5D4  C62:6D4  C62.121C23  C62:8D4  C2xS3xD12  D12:13D6  C2xS3xC3:D4
C2xC3:D12 is a maximal quotient of
 D12:18D6  D12.27D6  D12.28D6  D12.29D6  Dic6.29D6  D6:7Dic6  C12.27D12  C12.28D12  Dic3:Dic6  C12.30D12  C12:7D12  C12:D12  C12:2D12  C62.57D4  C62.60D4  C62:5D4  C62:6D4  C62:8D4

Matrix representation of C2xC3:D12 ►in GL6(Z)

100000
010000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
0000-11
0000-10
,
010000
-100000
00-1100
00-1000
000001
000010
,
-100000
010000
00-1000
00-1100
000001
000010

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

C2xC3:D12 in GAP, Magma, Sage, TeX 

C_2\times C_3\rtimes D_{12}
 
% in TeX
 
G:=Group("C2xC3:D12");
 
// GroupNames label
 
G:=SmallGroup(144,151);
 
// by ID
 
G=gap.SmallGroup(144,151);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,55,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C2xC3:D12 in TeX

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