metabelian, supersoluble, monomial
Aliases: C32⋊62+ 1+4, C62.148C23, C23⋊3S32, C3⋊D4⋊10D6, (C22×C6)⋊9D6, Dic3⋊D6⋊7C2, (C2×Dic3)⋊8D6, (C22×S3)⋊8D6, C3⋊4(D4⋊6D6), D6.4D6⋊7C2, D6.3D6⋊6C2, (C3×C6).37C24, C6.37(S3×C23), (C2×C62)⋊8C22, D6⋊S3⋊7C22, C3⋊D12⋊7C22, (S3×C6).18C23, (S3×Dic3)⋊4C22, D6.18(C22×S3), C6.D6⋊4C22, C32⋊2Q8⋊6C22, (C6×Dic3)⋊10C22, C32⋊7D4⋊10C22, C3⋊Dic3.28C23, Dic3.17(C22×S3), (C3×Dic3).19C23, (C2×S32)⋊7C22, (C6×C3⋊D4)⋊8C2, (S3×C3⋊D4)⋊6C2, (C2×C3⋊D4)⋊12S3, (S3×C2×C6)⋊13C22, C22.10(C2×S32), C2.37(C22×S32), (C2×C3⋊S3).31C23, (C2×C32⋊7D4)⋊17C2, (C3×C3⋊D4)⋊14C22, (C22×C3⋊S3)⋊10C22, (C2×C6).163(C22×S3), (C2×C3⋊Dic3)⋊13C22, SmallGroup(288,978)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊2+ 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, cac-1=dad=a-1, ae=ea, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >
Subgroups: 1410 in 359 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, 2+ 1+4, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×C62, D4⋊6D6, D6.3D6, D6.4D6, S3×C3⋊D4, Dic3⋊D6, C6×C3⋊D4, C2×C32⋊7D4, C32⋊2+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4⋊6D6, C22×S32, C32⋊2+ 1+4
►On 24 points - transitive group 24T582
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(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 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
(1 22 3 24)(2 23 4 21)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
G:=sub<Sym(24)| (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (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)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24), (1,22,3,24)(2,23,4,21)(5,15,7,13)(6,16,8,14)(9,19,11,17)(10,20,12,18), (5,7)(6,8)(9,11)(10,12)(21,23)(22,24)>;
G:=Group( (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (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)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24), (1,22,3,24)(2,23,4,21)(5,15,7,13)(6,16,8,14)(9,19,11,17)(10,20,12,18), (5,7)(6,8)(9,11)(10,12)(21,23)(22,24) );
G=PermutationGroup([[(1,16,17),(2,18,13),(3,14,19),(4,20,15),(5,23,10),(6,11,24),(7,21,12),(8,9,22)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(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),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24)], [(1,22,3,24),(2,23,4,21),(5,15,7,13),(6,16,8,14),(9,19,11,17),(10,20,12,18)], [(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)]])
G:=TransitiveGroup(24,582);
►On 24 points - transitive group 24T609
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 10 14)(6 15 11)(7 12 16)(8 13 9)
(1 21 20)(2 22 17)(3 23 18)(4 24 19)(5 14 10)(6 15 11)(7 16 12)(8 13 9)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
(1 2 3 4)(5 8 7 6)(9 16 11 14)(10 13 12 15)(17 23 19 21)(18 24 20 22)
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
G:=sub<Sym(24)| (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,10,14)(6,15,11)(7,12,16)(8,13,9), (1,21,20)(2,22,17)(3,23,18)(4,24,19)(5,14,10)(6,15,11)(7,16,12)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22), (1,2,3,4)(5,8,7,6)(9,16,11,14)(10,13,12,15)(17,23,19,21)(18,24,20,22), (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23)>;
G:=Group( (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,10,14)(6,15,11)(7,12,16)(8,13,9), (1,21,20)(2,22,17)(3,23,18)(4,24,19)(5,14,10)(6,15,11)(7,16,12)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22), (1,2,3,4)(5,8,7,6)(9,16,11,14)(10,13,12,15)(17,23,19,21)(18,24,20,22), (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23) );
G=PermutationGroup([[(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,10,14),(6,15,11),(7,12,16),(8,13,9)], [(1,21,20),(2,22,17),(3,23,18),(4,24,19),(5,14,10),(6,15,11),(7,16,12),(8,13,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)], [(1,2,3,4),(5,8,7,6),(9,16,11,14),(10,13,12,15),(17,23,19,21),(18,24,20,22)], [(1,5),(2,6),(3,7),(4,8),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)]])
G:=TransitiveGroup(24,609);
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6Q | 6R | 6S | 6T | 6U | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | D4⋊6D6 | C32⋊2+ 1+4 |
| kernel | C32⋊2+ 1+4 | D6.3D6 | D6.4D6 | S3×C3⋊D4 | Dic3⋊D6 | C6×C3⋊D4 | C2×C32⋊7D4 | C2×C3⋊D4 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C32 | C23 | C22 | C3 | C1 |
| # reps | 1 | 4 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 3 | 4 | 4 |
Matrix representation of C32⋊2+ 1+4 ►in GL4(𝔽7) generated by
| 4 | 1 | 2 | 1 |
| 3 | 3 | 6 | 4 |
| 6 | 6 | 4 | 2 |
| 6 | 1 | 4 | 1 |
| 6 | 0 | 3 | 1 |
| 5 | 3 | 6 | 0 |
| 1 | 6 | 4 | 5 |
| 3 | 3 | 2 | 6 |
| 4 | 6 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 3 | 2 | 1 | 4 |
| 0 | 4 | 3 | 6 |
| 1 | 4 | 2 | 0 |
| 5 | 3 | 3 | 1 |
| 4 | 6 | 0 | 5 |
| 3 | 3 | 2 | 3 |
| 4 | 3 | 3 | 4 |
| 3 | 6 | 4 | 1 |
| 3 | 0 | 3 | 4 |
| 0 | 3 | 4 | 1 |
| 3 | 0 | 3 | 1 |
| 5 | 0 | 6 | 0 |
| 1 | 6 | 1 | 5 |
| 3 | 3 | 2 | 3 |
G:=sub<GL(4,GF(7))| [4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,3,3,0,6,3,2,4,0,0,1,3,0,0,4,6],[1,5,4,3,4,3,6,3,2,3,0,2,0,1,5,3],[4,3,3,0,3,6,0,3,3,4,3,4,4,1,4,1],[3,5,1,3,0,0,6,3,3,6,1,2,1,0,5,3] >;
C32⋊2+ 1+4 in GAP, Magma, Sage, TeX
C_3^2\rtimes 2_+^{1+4} % in TeX
G:=Group("C3^2:ES+(2,2)"); // GroupNames label
G:=SmallGroup(288,978);
// by ID
G=gap.SmallGroup(288,978);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=d^2=f^2=1,e^2=c^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c^2*e>;
// generators/relations