metabelian, supersoluble, monomial
Aliases: C62.116C23, C62⋊8(C2×C4), C23.33S32, C6.70(S3×D4), (C2×Dic3)⋊12D6, (C22×C6).76D6, C2.5(Dic3⋊D6), C6.D4⋊11S3, C6.D12⋊16C2, (C6×Dic3)⋊14C22, C22⋊3(C6.D6), (C2×C62).35C22, (C2×C6)⋊8(C4×S3), C6.39(S3×C2×C4), C3⋊2(S3×C22⋊C4), (C22×C3⋊S3)⋊5C4, (C2×C3⋊S3).62D4, C22.57(C2×S32), C32⋊8(C2×C22⋊C4), C3⋊S3⋊3(C22⋊C4), (C3×C6).162(C2×D4), (C23×C3⋊S3).2C2, (C2×C6.D6)⋊14C2, (C3×C6).71(C22×C4), C2.16(C2×C6.D6), (C3×C6.D4)⋊20C2, (C2×C6).135(C22×S3), (C22×C3⋊S3).77C22, (C2×C3⋊S3)⋊15(C2×C4), SmallGroup(288,622)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.116C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >
Subgroups: 1538 in 331 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C6.D6, C6×Dic3, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, S3×C22⋊C4, C6.D12, C3×C6.D4, C2×C6.D6, C23×C3⋊S3, C62.116C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, S32, S3×C2×C4, S3×D4, C6.D6, C2×S32, S3×C22⋊C4, C2×C6.D6, Dic3⋊D6, C62.116C23
►On 24 points - transitive group 24T673
(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 17 5 15 3 13)(2 18 6 16 4 14)(7 19 9 21 11 23)(8 20 10 22 12 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)>;
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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20) );
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)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,19,9,21,11,23),(8,20,10,22,12,24)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,18,10,15),(8,17,11,14),(9,16,12,13)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)]])
G:=TransitiveGroup(24,673);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 4A | ··· | 4H | 6A | ··· | 6F | 6G | ··· | 6Q | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | 2 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4×S3 | S32 | S3×D4 | C6.D6 | C2×S32 | Dic3⋊D6 |
| kernel | C62.116C23 | C6.D12 | C3×C6.D4 | C2×C6.D6 | C23×C3⋊S3 | C22×C3⋊S3 | C6.D4 | C2×C3⋊S3 | C2×Dic3 | C22×C6 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 | 4 | 2 | 8 | 1 | 4 | 2 | 1 | 4 |
Matrix representation of C62.116C23 ►in GL6(𝔽13)
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,5,0,0,0,0,10,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.116C23 in GAP, Magma, Sage, TeX
C_6^2._{116}C_2^3 % in TeX
G:=Group("C6^2.116C2^3"); // GroupNames label
G:=SmallGroup(288,622);
// by ID
G=gap.SmallGroup(288,622);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=b^3*d>;
// generators/relations