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G = C12.31D6  order 144 = 24·32

5th non-split extension by C12 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C12.31D6, C32⋊3M4(2), C3⋊C8⋊5S3, C4.16S32, C6.2(C4×S3), C3⋊1(C8⋊S3), C3⋊Dic3.3C4, (C3×C12).30C22, C2.3(C6.D6), (C3×C3⋊C8)⋊8C2, (C2×C3⋊S3).3C4, (C4×C3⋊S3).4C2, (C3×C6).11(C2×C4), SmallGroup(144,55)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C12.31D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.31D6
Lower central
C32 — C3×C6 — C12.31D6
Upper central
C1 — C4

Generators and relations for C12.31D6
 G = < a,b,c | a12=c2=1, b6=a3, bab-1=cac=a5, cbc=b5 >

C1
C2
18C2
C3
C3
2C3
C4
9C4
9C22
C6
C6
2C6
6S3
6S3
6S3
6S3
C32
3C8
3C8
9C2×C4
C12
C12
2C12
3Dic3
3Dic3
3D6
3D6
6Dic3
6D6
C3×C6
2C3⋊S3
9M4(2)
C3⋊C8
C3⋊C8
3C24
3C24
3C4×S3
3C4×S3
6C4×S3
C3⋊Dic3
C2×C3⋊S3
C3×C12
3C8⋊S3
3C8⋊S3
C3×C3⋊C8
C4×C3⋊S3
C3×C3⋊C8
C12.31D6

Character table of C12.31D6

 class 12A2B3A3B3C4A4B4C6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D24E24F24G24H
 size 11182241118224666622224466666666
ρ1111111111111111111111111111111    trivial
ρ211-111111-11111-11-11111111-1-11-1-111    linear of order 2
ρ311-111111-1111-11-11111111-111-111-1-1    linear of order 2
ρ4111111111111-1-1-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1111-iii-i-1-1-1-1-1-1i-i-iiii-i-i    linear of order 4
ρ611-1111-1-11111-i-iii-1-1-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ711-1111-1-11111ii-i-i-1-1-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ8111111-1-1-1111i-i-ii-1-1-1-1-1-1-iii-i-i-iii    linear of order 4
ρ92202-1-1220-12-120202-12-1-1-1-100-100-1-1    orthogonal lifted from S3
ρ10220-12-12202-1-10-20-2-12-12-1-101101100    orthogonal lifted from D6
ρ11220-12-12202-1-10202-12-12-1-10-1-10-1-100    orthogonal lifted from S3
ρ122202-1-1220-12-1-20-202-12-1-1-110010011    orthogonal lifted from D6
ρ132-202222i-2i0-2-2-200002i-2i-2i2i-2i2i00000000    complex lifted from M4(2)
ρ142202-1-1-2-20-12-12i0-2i0-21-2111i00i00-i-i    complex lifted from C4×S3
ρ15220-12-1-2-202-1-10-2i02i1-21-2110-i-i0ii00    complex lifted from C4×S3
ρ162202-1-1-2-20-12-1-2i02i0-21-2111-i00-i00ii    complex lifted from C4×S3
ρ17220-12-1-2-202-1-102i0-2i1-21-2110ii0-i-i00    complex lifted from C4×S3
ρ182-20222-2i2i0-2-2-20000-2i2i2i-2i2i-2i00000000    complex lifted from M4(2)
ρ192-20-12-1-2i2i0-2110000i2i-i-2i-ii02ζ8ζ3+ζ82ζ85ζ3+ζ8502ζ83ζ3+ζ832ζ87ζ3+ζ8700    complex lifted from C8⋊S3
ρ202-20-12-12i-2i0-2110000-i-2ii2ii-i02ζ83ζ3+ζ832ζ87ζ3+ζ8702ζ8ζ3+ζ82ζ85ζ3+ζ8500    complex lifted from C8⋊S3
ρ212-20-12-1-2i2i0-2110000i2i-i-2i-ii02ζ85ζ3+ζ852ζ8ζ3+ζ802ζ87ζ3+ζ872ζ83ζ3+ζ8300    complex lifted from C8⋊S3
ρ222-20-12-12i-2i0-2110000-i-2ii2ii-i02ζ87ζ3+ζ872ζ83ζ3+ζ8302ζ85ζ3+ζ852ζ8ζ3+ζ800    complex lifted from C8⋊S3
ρ232-202-1-1-2i2i01-210000-2i-i2ii-ii2ζ8ζ3+ζ8002ζ85ζ3+ζ85002ζ87ζ3+ζ872ζ83ζ3+ζ83    complex lifted from C8⋊S3
ρ242-202-1-1-2i2i01-210000-2i-i2ii-ii2ζ85ζ3+ζ85002ζ8ζ3+ζ8002ζ83ζ3+ζ832ζ87ζ3+ζ87    complex lifted from C8⋊S3
ρ252-202-1-12i-2i01-2100002ii-2i-ii-i2ζ87ζ3+ζ87002ζ83ζ3+ζ83002ζ8ζ3+ζ82ζ85ζ3+ζ85    complex lifted from C8⋊S3
ρ262-202-1-12i-2i01-2100002ii-2i-ii-i2ζ83ζ3+ζ83002ζ87ζ3+ζ87002ζ85ζ3+ζ852ζ8ζ3+ζ8    complex lifted from C8⋊S3
ρ27440-2-21-4-40-2-2100002222-1-100000000    orthogonal lifted from C6.D6
ρ28440-2-21440-2-210000-2-2-2-21100000000    orthogonal lifted from S32
ρ294-40-2-214i-4i022-10000-2i2i2i-2i-ii00000000    complex faithful
ρ304-40-2-21-4i4i022-100002i-2i-2i2ii-i00000000    complex faithful

Permutation representations of C12.31D6
►On 24 points - transitive group 24T236
Generators in S24
(1 3 5 7 9 11 13 15 17 19 21 23)(2 12 22 8 18 4 14 24 10 20 6 16)

(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 21)(3 7)(4 12)(5 17)(6 22)(9 13)(10 18)(11 23)(15 19)(16 24)
 
G:=sub<Sym(24)| (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(3,7)(4,12)(5,17)(6,22)(9,13)(10,18)(11,23)(15,19)(16,24)>;
 
G:=Group( (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(3,7)(4,12)(5,17)(6,22)(9,13)(10,18)(11,23)(15,19)(16,24) );
 
G=PermutationGroup([[(1,3,5,7,9,11,13,15,17,19,21,23),(2,12,22,8,18,4,14,24,10,20,6,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,21),(3,7),(4,12),(5,17),(6,22),(9,13),(10,18),(11,23),(15,19),(16,24)]])
 
G:=TransitiveGroup(24,236);
 

C12.31D6 is a maximal subgroup of
 C4.S3≀C2  (C3×C12).D4  C32⋊C4≀C2  C4.19S3≀C2  S3×C8⋊S3  C24.63D6  C24.D6  C3⋊C8.22D6  C3⋊C8⋊20D6  D12.D6  Dic6.D6  D12⋊5D6  D12.10D6  Dic6.10D6  D12.15D6  C36.40D6  He3⋊M4(2)  C33⋊9M4(2)  C33⋊10M4(2)
C12.31D6 is a maximal quotient of
 C2.Dic32  C12.78D12  C12.15Dic6  C36.40D6  He3⋊3M4(2)  C33⋊9M4(2)  C33⋊10M4(2)

Matrix representation of C12.31D6 ►in GL4(𝔽5) generated by

0010
0004
1030
0403
,
0003
3000
0300
1030
,
4000
0002
2010
0300
G:=sub<GL(4,GF(5))| [0,0,1,0,0,0,0,4,1,0,3,0,0,4,0,3],[0,3,0,1,0,0,3,0,0,0,0,3,3,0,0,0],[4,0,2,0,0,0,0,3,0,0,1,0,0,2,0,0] >;
 

C12.31D6 in GAP, Magma, Sage, TeX 

C_{12}._{31}D_6
 
% in TeX
 
G:=Group("C12.31D6");
 
// GroupNames label
 
G:=SmallGroup(144,55);
 
// by ID
 
G=gap.SmallGroup(144,55);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,31,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=c^2=1,b^6=a^3,b*a*b^-1=c*a*c=a^5,c*b*c=b^5>;
 
// generators/relations
 

Export

Subgroup lattice of C12.31D6 in TeX
Character table of C12.31D6 in TeX

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