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G = C2×C3⋊D12  order 144 = 24·32

Direct product of C2 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C3⋊D12, D6⋊5D6, C6⋊2D12, Dic3⋊4D6, C62.11C22, (C3×C6)⋊3D4, C3⋊3(C2×D12), C32⋊5(C2×D4), C6⋊1(C3⋊D4), C22.11S32, (C2×C6).16D6, (C22×S3)⋊3S3, (S3×C6)⋊6C22, (C6×Dic3)⋊5C2, (C2×Dic3)⋊4S3, (C3×C6).15C23, C6.15(C22×S3), (C3×Dic3)⋊4C22, (S3×C2×C6)⋊2C2, C2.15(C2×S32), C3⋊1(C2×C3⋊D4), (C2×C3⋊S3)⋊3C22, (C22×C3⋊S3)⋊1C2, SmallGroup(144,151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×C3⋊D12
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12 — C2×C3⋊D12
Lower central
C32 — C3×C6 — C2×C3⋊D12
Upper central
C1 — C22

Generators and relations for C2×C3⋊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, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C3×Dic3, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C2×C3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C2×C3⋊D12

Character table of C2×C3⋊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 C2×S32
ρ3044-4-40000-2-2100-2-22222-11-100000000    orthogonal lifted from C3⋊D12

Permutation representations of C2×C3⋊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);
 

C2×C3⋊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  C2×S3×D12  D12⋊13D6  C2×S3×C3⋊D4
C2×C3⋊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 C2×C3⋊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] >;
 

C2×C3⋊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 C2×C3⋊D12 in TeX

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