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G = C3×D12order 72 = 23·32

Direct product of C3 and D12

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D12, C121C6, C123S3, D61C6, C324D4, C6.19D6, C4⋊(C3×S3), C31(C3×D4), (S3×C6)⋊3C2, (C3×C12)⋊2C2, C2.4(S3×C6), C6.3(C2×C6), (C3×C6).8C22, SmallGroup(72,28)

Series: Derived Chief Lower central Upper central

C1C6 — C3×D12
C1C3C6C3×C6S3×C6 — C3×D12
C3C6 — C3×D12
C1C6C12

Generators and relations for C3×D12
 G = < a,b,c | a3=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

6C2
6C2
2C3
3C22
3C22
2C6
2S3
2S3
6C6
6C6
3D4
2C12
3C2×C6
3C2×C6
2C3×S3
2C3×S3
3C3×D4

Character table of C3×D12

 class 12A2B2C3A3B3C3D3E46A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 116611222211222666622222222
ρ1111111111111111111111111111    trivial
ρ211-1111111-111111-11-11-1-1-1-1-1-1-1-1    linear of order 2
ρ3111-111111-1111111-11-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411-1-111111111111-1-1-1-111111111    linear of order 2
ρ5111-1ζ32ζ3ζ3ζ321-1ζ3ζ32ζ32ζ31ζ3ζ6ζ32ζ65ζ65ζ65-1-1ζ6ζ6ζ6ζ65    linear of order 6
ρ6111-1ζ3ζ32ζ32ζ31-1ζ32ζ3ζ3ζ321ζ32ζ65ζ3ζ6ζ6ζ6-1-1ζ65ζ65ζ65ζ6    linear of order 6
ρ711-11ζ3ζ32ζ32ζ31-1ζ32ζ3ζ3ζ321ζ6ζ3ζ65ζ32ζ6ζ6-1-1ζ65ζ65ζ65ζ6    linear of order 6
ρ811-1-1ζ3ζ32ζ32ζ311ζ32ζ3ζ3ζ321ζ6ζ65ζ65ζ6ζ32ζ3211ζ3ζ3ζ3ζ32    linear of order 6
ρ91111ζ3ζ32ζ32ζ311ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ3211ζ3ζ3ζ3ζ32    linear of order 3
ρ101111ζ32ζ3ζ3ζ3211ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ311ζ32ζ32ζ32ζ3    linear of order 3
ρ1111-1-1ζ32ζ3ζ3ζ3211ζ3ζ32ζ32ζ31ζ65ζ6ζ6ζ65ζ3ζ311ζ32ζ32ζ32ζ3    linear of order 6
ρ1211-11ζ32ζ3ζ3ζ321-1ζ3ζ32ζ32ζ31ζ65ζ32ζ6ζ3ζ65ζ65-1-1ζ6ζ6ζ6ζ65    linear of order 6
ρ13220022-1-1-1222-1-1-10000-1-1-1-12-1-12    orthogonal lifted from S3
ρ14220022-1-1-1-222-1-1-100001111-211-2    orthogonal lifted from D6
ρ152-200222220-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ162-20022-1-1-10-2-211100003-33-303-30    orthogonal lifted from D12
ρ172-20022-1-1-10-2-21110000-33-330-330    orthogonal lifted from D12
ρ182200-1+-3-1--3ζ6ζ65-1-2-1--3-1+-3ζ65ζ6-10000ζ32ζ32111--3ζ3ζ31+-3    complex lifted from S3×C6
ρ192-200-1+-3-1--3-1--3-1+-3201+-31--31--31+-3-2000000000000    complex lifted from C3×D4
ρ202200-1+-3-1--3ζ6ζ65-12-1--3-1+-3ζ65ζ6-10000ζ6ζ6-1-1-1+-3ζ65ζ65-1--3    complex lifted from C3×S3
ρ212200-1--3-1+-3ζ65ζ6-1-2-1+-3-1--3ζ6ζ65-10000ζ3ζ3111+-3ζ32ζ321--3    complex lifted from S3×C6
ρ222-200-1+-3-1--3ζ6ζ65-101+-31--3ζ3ζ3210000ζ4ζ32+2ζ4ζ43ζ32+2ζ43-330ζ43ζ3+2ζ43ζ4ζ3+2ζ40    complex faithful
ρ232-200-1--3-1+-3ζ65ζ6-101--31+-3ζ32ζ310000ζ4ζ3+2ζ4ζ43ζ3+2ζ433-30ζ43ζ32+2ζ43ζ4ζ32+2ζ40    complex faithful
ρ242-200-1--3-1+-3-1+-3-1--3201--31+-31+-31--3-2000000000000    complex lifted from C3×D4
ρ252200-1--3-1+-3ζ65ζ6-12-1+-3-1--3ζ6ζ65-10000ζ65ζ65-1-1-1--3ζ6ζ6-1+-3    complex lifted from C3×S3
ρ262-200-1--3-1+-3ζ65ζ6-101--31+-3ζ32ζ310000ζ43ζ3+2ζ43ζ4ζ3+2ζ4-330ζ4ζ32+2ζ4ζ43ζ32+2ζ430    complex faithful
ρ272-200-1+-3-1--3ζ6ζ65-101+-31--3ζ3ζ3210000ζ43ζ32+2ζ43ζ4ζ32+2ζ43-30ζ4ζ3+2ζ4ζ43ζ3+2ζ430    complex faithful

Permutation representations of C3×D12
On 24 points - transitive group 24T67
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

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

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

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

G:=TransitiveGroup(24,67);

C3×D12 is a maximal subgroup of
C322D8  C3⋊D24  Dic6⋊S3  D12.S3  D125S3  D12⋊S3  D6⋊D6  C3×S3×D4  He34D4  D36⋊C3  He35D4  D12.A4
C3×D12 is a maximal quotient of
He34D4  D36⋊C3

Matrix representation of C3×D12 in GL2(𝔽13) generated by

90
09
,
60
411
,
29
411
G:=sub<GL(2,GF(13))| [9,0,0,9],[6,4,0,11],[2,4,9,11] >;

C3×D12 in GAP, Magma, Sage, TeX

C_3\times D_{12}
% in TeX

G:=Group("C3xD12");
// GroupNames label

G:=SmallGroup(72,28);
// by ID

G=gap.SmallGroup(72,28);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,141,66,1204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D12 in TeX
Character table of C3×D12 in TeX

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