metabelian, supersoluble, monomial
Aliases: C62.96D6, C32⋊7D4⋊7S3, C33⋊9D4⋊5C2, C33⋊5Q8⋊8C2, C33⋊26(C4○D4), C3⋊Dic3.22D6, C3⋊7(D6.3D6), C3⋊5(D6.4D6), C32⋊18(C4○D12), (C32×C6).71C23, (C3×C62).34C22, C22.(C32⋊4D6), C32⋊18(D4⋊2S3), (C2×C6).12S32, C6.100(C2×S32), C33⋊9(C2×C4)⋊5C2, (C2×C3⋊S3).20D6, (C2×C3⋊Dic3)⋊13S3, (C6×C3⋊Dic3)⋊11C2, (C3×C32⋊7D4)⋊5C2, (C6×C3⋊S3).31C22, C2.7(C2×C32⋊4D6), (C3×C6).121(C22×S3), (C3×C3⋊Dic3).25C22, SmallGroup(432,693)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.96D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=a-1, dad-1=ab3, cbc-1=dbd-1=b-1, dcd-1=c5 >
Subgroups: 1032 in 218 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, D4⋊2S3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C3×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, D6.3D6, D6.4D6, C33⋊9(C2×C4), C33⋊9D4, C33⋊5Q8, C6×C3⋊Dic3, C3×C32⋊7D4, C62.96D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, C32⋊4D6, D6.3D6, D6.4D6, C2×C32⋊4D6, C62.96D6
►On 24 points - transitive group 24T1285
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 15 17 19 21 23)(14 24 22 20 18 16)
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 24 22 20 18 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 15 7 21)(2 20 8 14)(3 13 9 19)(4 18 10 24)(5 23 11 17)(6 16 12 22)
G:=sub<Sym(24)| (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15,7,21)(2,20,8,14)(3,13,9,19)(4,18,10,24)(5,23,11,17)(6,16,12,22)>;
G:=Group( (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15,7,21)(2,20,8,14)(3,13,9,19)(4,18,10,24)(5,23,11,17)(6,16,12,22) );
G=PermutationGroup([[(1,5,9),(2,10,6),(3,7,11),(4,12,8),(13,15,17,19,21,23),(14,24,22,20,18,16)], [(1,3,5,7,9,11),(2,12,10,8,6,4),(13,15,17,19,21,23),(14,24,22,20,18,16)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15,7,21),(2,20,8,14),(3,13,9,19),(4,18,10,24),(5,23,11,17),(6,16,12,22)]])
G:=TransitiveGroup(24,1285);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6E | 6F | ··· | 6V | 6W | 6X | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 9 | 9 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 36 | 36 | 18 | 18 | 18 | 18 | 36 | 36 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | C32⋊4D6 | D6.3D6 | D6.4D6 | C2×C32⋊4D6 | C62.96D6 |
| kernel | C62.96D6 | C33⋊9(C2×C4) | C33⋊9D4 | C33⋊5Q8 | C6×C3⋊Dic3 | C3×C32⋊7D4 | C2×C3⋊Dic3 | C32⋊7D4 | C3⋊Dic3 | C2×C3⋊S3 | C62 | C33 | C32 | C2×C6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 3 | 2 | 4 | 3 | 2 | 3 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C62.96D6 ►in GL4(𝔽7) generated by
| 3 | 0 | 1 | 1 |
| 3 | 6 | 2 | 2 |
| 1 | 1 | 1 | 5 |
| 0 | 0 | 0 | 4 |
| 1 | 5 | 2 | 6 |
| 0 | 3 | 0 | 2 |
| 3 | 3 | 0 | 1 |
| 0 | 0 | 0 | 5 |
| 0 | 6 | 6 | 2 |
| 6 | 6 | 5 | 0 |
| 1 | 6 | 3 | 6 |
| 1 | 1 | 3 | 5 |
| 3 | 2 | 3 | 1 |
| 1 | 0 | 1 | 0 |
| 4 | 4 | 4 | 6 |
| 4 | 3 | 5 | 0 |
G:=sub<GL(4,GF(7))| [3,3,1,0,0,6,1,0,1,2,1,0,1,2,5,4],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[0,6,1,1,6,6,6,1,6,5,3,3,2,0,6,5],[3,1,4,4,2,0,4,3,3,1,4,5,1,0,6,0] >;
C62.96D6 in GAP, Magma, Sage, TeX
C_6^2._{96}D_6 % in TeX
G:=Group("C6^2.96D6"); // GroupNames label
G:=SmallGroup(432,693);
// by ID
G=gap.SmallGroup(432,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations