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G = C2×C3⋊D12order 144 = 24·32

Direct product of C2 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C3⋊D12, D65D6, C62D12, Dic34D6, C62.11C22, (C3×C6)⋊3D4, C33(C2×D12), C325(C2×D4), C61(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), C31(C2×C3⋊D4), (C2×C3⋊S3)⋊3C22, (C22×C3⋊S3)⋊1C2, SmallGroup(144,151)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C3⋊D12
C1C3C32C3×C6S3×C6C3⋊D12 — C2×C3⋊D12
C32C3×C6 — C2×C3⋊D12
C1C22

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 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×10], C6 [×2], C6 [×4], C6 [×5], C2×C4, D4 [×4], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×2], D6 [×16], C2×C6 [×2], C2×C6 [×5], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×4], C2×C12, C22×S3, C22×S3 [×3], C22×C6, C3×Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×D12, C2×C3⋊D4, C3⋊D12 [×4], C6×Dic3, S3×C2×C6, C22×C3⋊S3, C2×C3⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], 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
ρ1922-2-20000-12-100-12-211-21-110000-33-33    orthogonal lifted from D12
ρ202-22-20000-12-1001-22-11-211-10000-333-3    orthogonal lifted from D12
ρ2122-2-20000-12-100-12-211-21-1100003-33-3    orthogonal lifted from D12
ρ222-22-20000-12-1001-22-11-211-100003-3-33    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 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 21 17)(14 18 22)(15 23 19)(16 20 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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)

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

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

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

G:=TransitiveGroup(24,230);

C2×C3⋊D12 is a maximal subgroup of
C62.20C23  Dic3.D12  C62.23C23  C62.24C23  Dic34D12  C62.51C23  Dic3⋊D12  D6.D12  C62.74C23  D6⋊D12  C62.77C23  C127D12  Dic33D12  C12⋊D12  C62.82C23  C122D12  D64D12  D65D12  C62.100C23  C62.113C23  C625D4  C626D4  C62.121C23  C628D4  C2×S3×D12  D1213D6  C2×S3×C3⋊D4
C2×C3⋊D12 is a maximal quotient of
D1218D6  D12.27D6  D12.28D6  D12.29D6  Dic6.29D6  D67Dic6  C12.27D12  C12.28D12  Dic3⋊Dic6  C12.30D12  C127D12  C12⋊D12  C122D12  C62.57D4  C62.60D4  C625D4  C626D4  C628D4

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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