metabelian, supersoluble, monomial
Aliases: C62.31D4, C23.8S32, (C2×C6).9D12, (C2×Dic3)⋊Dic3, (C6×Dic3)⋊1C4, C6.41(D6⋊C4), C62⋊5C4⋊1C2, C6.D4⋊1S3, C32⋊4(C23⋊C4), C62.34(C2×C4), (C22×C6).54D6, (C22×S3)⋊2Dic3, (C2×C62).7C22, C22.5(S3×Dic3), C2.11(D6⋊Dic3), C3⋊3(C23.6D6), C3⋊1(C23.7D6), C6.11(C6.D4), C22.3(D6⋊S3), C22.8(C3⋊D12), (S3×C2×C6)⋊1C4, (C2×C6).70(C4×S3), (C2×C3⋊D4).1S3, (C6×C3⋊D4).4C2, (C2×C6).6(C2×Dic3), (C3×C6.D4)⋊1C2, (C2×C6).12(C3⋊D4), (C3×C6).39(C22⋊C4), SmallGroup(288,228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.31D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=a3c-1 >
Subgroups: 474 in 121 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C23⋊C4, C3×Dic3, C3⋊Dic3, S3×C6, C62, C62, C6.D4, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.6D6, C23.7D6, C3×C6.D4, C62⋊5C4, C6×C3⋊D4, C62.31D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, C23⋊C4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C23.6D6, C23.7D6, D6⋊Dic3, C62.31D4
►On 24 points - transitive group 24T581
(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 4 2 5 3 6)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 19 5 22)(2 21 6 24)(3 23 4 20)(7 13)(8 15)(9 17)(10 16)(11 18)(12 14)
(1 13)(2 17)(3 15)(4 18)(5 16)(6 14)(7 22)(8 20)(9 24)(10 19)(11 23)(12 21)
G:=sub<Sym(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), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,19,5,22)(2,21,6,24)(3,23,4,20)(7,13)(8,15)(9,17)(10,16)(11,18)(12,14), (1,13)(2,17)(3,15)(4,18)(5,16)(6,14)(7,22)(8,20)(9,24)(10,19)(11,23)(12,21)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,19,5,22)(2,21,6,24)(3,23,4,20)(7,13)(8,15)(9,17)(10,16)(11,18)(12,14), (1,13)(2,17)(3,15)(4,18)(5,16)(6,14)(7,22)(8,20)(9,24)(10,19)(11,23)(12,21) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,4,2,5,3,6),(7,11,9,10,8,12),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,19,5,22),(2,21,6,24),(3,23,4,20),(7,13),(8,15),(9,17),(10,16),(11,18),(12,14)], [(1,13),(2,17),(3,15),(4,18),(5,16),(6,14),(7,22),(8,20),(9,24),(10,19),(11,23),(12,21)]])
G:=TransitiveGroup(24,581);
►On 24 points - transitive group 24T614
(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 5 14 3 18)(2 17 6 15 4 13)(7 20 9 22 11 24)(8 21 10 23 12 19)
(2 13)(3 5)(4 17)(6 15)(7 10 22 19)(8 24 23 9)(11 12 20 21)(16 18)
(1 7)(2 23)(3 9)(4 19)(5 11)(6 21)(8 15)(10 17)(12 13)(14 22)(16 24)(18 20)
G:=sub<Sym(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), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,5,14,3,18),(2,17,6,15,4,13),(7,20,9,22,11,24),(8,21,10,23,12,19)], [(2,13),(3,5),(4,17),(6,15),(7,10,22,19),(8,24,23,9),(11,12,20,21),(16,18)], [(1,7),(2,23),(3,9),(4,19),(5,11),(6,21),(8,15),(10,17),(12,13),(14,22),(16,24),(18,20)]])
G:=TransitiveGroup(24,614);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6F | 6G | ··· | 6Q | 6R | 6S | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 2 | 2 | 4 | 12 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | ··· | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | S3 | D4 | Dic3 | Dic3 | D6 | C4×S3 | D12 | C3⋊D4 | C23⋊C4 | S32 | S3×Dic3 | D6⋊S3 | C3⋊D12 | C23.6D6 | C23.7D6 | C62.31D4 |
| kernel | C62.31D4 | C3×C6.D4 | C62⋊5C4 | C6×C3⋊D4 | C6×Dic3 | S3×C2×C6 | C6.D4 | C2×C3⋊D4 | C62 | C2×Dic3 | C22×S3 | C22×C6 | C2×C6 | C2×C6 | C2×C6 | C32 | C23 | C22 | C22 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C62.31D4 ►in GL4(𝔽7) generated by
| 5 | 2 | 3 | 2 |
| 1 | 4 | 3 | 4 |
| 2 | 2 | 3 | 3 |
| 0 | 0 | 0 | 2 |
| 1 | 5 | 2 | 6 |
| 0 | 3 | 0 | 2 |
| 3 | 3 | 0 | 1 |
| 0 | 0 | 0 | 5 |
| 2 | 0 | 5 | 2 |
| 4 | 2 | 0 | 4 |
| 3 | 3 | 3 | 1 |
| 1 | 6 | 3 | 0 |
| 4 | 1 | 4 | 5 |
| 2 | 2 | 0 | 4 |
| 3 | 4 | 2 | 4 |
| 4 | 4 | 5 | 6 |
G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[2,4,3,1,0,2,3,6,5,0,3,3,2,4,1,0],[4,2,3,4,1,2,4,4,4,0,2,5,5,4,4,6] >;
C62.31D4 in GAP, Magma, Sage, TeX
C_6^2._{31}D_4 % in TeX
G:=Group("C6^2.31D4"); // GroupNames label
G:=SmallGroup(288,228);
// by ID
G=gap.SmallGroup(288,228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;
// generators/relations