Aliases: C24.6A4, C4⋊1D4⋊C6, C42⋊(C2×C6), C42⋊C6⋊C2, C42⋊2C2⋊C6, C42⋊C3⋊2C22, C23.5(C2×A4), C23.A4⋊1C2, C22.54C24⋊C3, C22.5(C22×A4), SmallGroup(192,1008)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C24.6A4 |
Subgroups: 342 in 67 conjugacy classes, 16 normal (10 characteristic)
C1, C2 [×5], C3, C4 [×3], C22, C22 [×8], C6 [×3], C2×C4 [×4], D4 [×4], C23, C23 [×2], C23 [×2], A4, C2×C6, C42, C22⋊C4 [×4], C4⋊C4 [×2], C22×C4, C2×D4 [×4], C24, C2×A4 [×3], C22≀C2, C4⋊D4 [×2], C22.D4, C42⋊2C2 [×2], C4⋊1D4, C42⋊C3, C22×A4, C22.54C24, C42⋊C6 [×2], C23.A4, C24.6A4
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C24.6A4
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ag=ga, ebe-1=bc=cb, fbf-1=bd=db, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >
►On 16 points - transitive group 16T420
Generators in S16
(2 4)(5 12)(6 11)(7 10)(8 9)(14 16)
(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)
G:=sub<Sym(16)| (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (2,14)(4,16)(5,10)(6,8)(7,12)(9,11), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12)>;
G:=Group( (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (2,14)(4,16)(5,10)(6,8)(7,12)(9,11), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12) );
G=PermutationGroup([(2,4),(5,12),(6,11),(7,10),(8,9),(14,16)], [(2,14),(4,16),(5,10),(6,8),(7,12),(9,11)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,13,5),(2,9,14,6),(3,10,15,7),(4,11,16,8)], [(2,10,6),(3,13,15),(4,7,11),(5,8,14),(9,16,12)])
G:=TransitiveGroup(16,420);
►On 24 points - transitive group 24T368
Generators in S24
(1 7)(2 8)(10 12)(14 16)(17 19)(22 24)
(1 7)(4 5)(10 12)(18 20)(21 23)(22 24)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(17 19)(18 20)
(1 7)(2 8)(3 6)(4 5)(13 15)(14 16)(21 23)(22 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 6 7 3)(2 5 8 4)(9 19 11 17)(10 20 12 18)(13 23)(14 24)(15 21)(16 22)
(1 24 12)(2 16 17)(3 13 11)(4 21 20)(5 23 18)(6 15 9)(7 22 10)(8 14 19)
G:=sub<Sym(24)| (1,7)(2,8)(10,12)(14,16)(17,19)(22,24), (1,7)(4,5)(10,12)(18,20)(21,23)(22,24), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(17,19)(18,20), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,7,3)(2,5,8,4)(9,19,11,17)(10,20,12,18)(13,23)(14,24)(15,21)(16,22), (1,24,12)(2,16,17)(3,13,11)(4,21,20)(5,23,18)(6,15,9)(7,22,10)(8,14,19)>;
G:=Group( (1,7)(2,8)(10,12)(14,16)(17,19)(22,24), (1,7)(4,5)(10,12)(18,20)(21,23)(22,24), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(17,19)(18,20), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,7,3)(2,5,8,4)(9,19,11,17)(10,20,12,18)(13,23)(14,24)(15,21)(16,22), (1,24,12)(2,16,17)(3,13,11)(4,21,20)(5,23,18)(6,15,9)(7,22,10)(8,14,19) );
G=PermutationGroup([(1,7),(2,8),(10,12),(14,16),(17,19),(22,24)], [(1,7),(4,5),(10,12),(18,20),(21,23),(22,24)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(17,19),(18,20)], [(1,7),(2,8),(3,6),(4,5),(13,15),(14,16),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6,7,3),(2,5,8,4),(9,19,11,17),(10,20,12,18),(13,23),(14,24),(15,21),(16,22)], [(1,24,12),(2,16,17),(3,13,11),(4,21,20),(5,23,18),(6,15,9),(7,22,10),(8,14,19)])
G:=TransitiveGroup(24,368);
►On 24 points - transitive group 24T373
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 12)(10 11)(13 14)(15 16)(17 21)(18 24)(19 23)(20 22)
(1 5)(4 7)(9 11)(10 12)(17 19)(21 23)
(1 5)(2 6)(3 8)(4 7)(17 19)(18 20)(21 23)(22 24)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 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 8 5 3)(2 7 6 4)(9 14)(10 15)(11 16)(12 13)(17 23 19 21)(18 24 20 22)
(1 10 19)(2 13 24)(3 14 18)(4 11 23)(5 12 17)(6 15 22)(7 9 21)(8 16 20)
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,23)(20,22), (1,5)(4,7)(9,11)(10,12)(17,19)(21,23), (1,5)(2,6)(3,8)(4,7)(17,19)(18,20)(21,23)(22,24), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,5,3)(2,7,6,4)(9,14)(10,15)(11,16)(12,13)(17,23,19,21)(18,24,20,22), (1,10,19)(2,13,24)(3,14,18)(4,11,23)(5,12,17)(6,15,22)(7,9,21)(8,16,20)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,23)(20,22), (1,5)(4,7)(9,11)(10,12)(17,19)(21,23), (1,5)(2,6)(3,8)(4,7)(17,19)(18,20)(21,23)(22,24), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,5,3)(2,7,6,4)(9,14)(10,15)(11,16)(12,13)(17,23,19,21)(18,24,20,22), (1,10,19)(2,13,24)(3,14,18)(4,11,23)(5,12,17)(6,15,22)(7,9,21)(8,16,20) );
G=PermutationGroup([(1,4),(2,3),(5,7),(6,8),(9,12),(10,11),(13,14),(15,16),(17,21),(18,24),(19,23),(20,22)], [(1,5),(4,7),(9,11),(10,12),(17,19),(21,23)], [(1,5),(2,6),(3,8),(4,7),(17,19),(18,20),(21,23),(22,24)], [(1,5),(2,6),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8,5,3),(2,7,6,4),(9,14),(10,15),(11,16),(12,13),(17,23,19,21),(18,24,20,22)], [(1,10,19),(2,13,24),(3,14,18),(4,11,23),(5,12,17),(6,15,22),(7,9,21),(8,16,20)])
G:=TransitiveGroup(24,373);
►On 24 points - transitive group 24T377
Generators in S24
(3 4)(5 6)(10 12)(13 15)(18 20)(22 24)
(1 7)(2 8)(3 6)(4 5)(9 21)(10 22)(11 23)(12 24)(13 20)(14 19)(15 18)(16 17)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(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 7 6 8)(9 11)(10 12)(13 14 15 16)(17 20 19 18)
(1 9 16)(2 11 14)(3 10 15)(4 12 13)(5 24 20)(6 22 18)(7 21 17)(8 23 19)
G:=sub<Sym(24)| (3,4)(5,6)(10,12)(13,15)(18,20)(22,24), (1,7)(2,8)(3,6)(4,5)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,24,20)(6,22,18)(7,21,17)(8,23,19)>;
G:=Group( (3,4)(5,6)(10,12)(13,15)(18,20)(22,24), (1,7)(2,8)(3,6)(4,5)(9,21)(10,22)(11,23)(12,24)(13,20)(14,19)(15,18)(16,17), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,9,16)(2,11,14)(3,10,15)(4,12,13)(5,24,20)(6,22,18)(7,21,17)(8,23,19) );
G=PermutationGroup([(3,4),(5,6),(10,12),(13,15),(18,20),(22,24)], [(1,7),(2,8),(3,6),(4,5),(9,21),(10,22),(11,23),(12,24),(13,20),(14,19),(15,18),(16,17)], [(1,2),(3,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,20,19,18)], [(1,9,16),(2,11,14),(3,10,15),(4,12,13),(5,24,20),(6,22,18),(7,21,17),(8,23,19)])
G:=TransitiveGroup(24,377);
►On 24 points - transitive group 24T380
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 12)(10 11)(13 16)(14 15)(17 18)(19 20)(21 22)(23 24)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 19)(14 18)(15 17)(16 20)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(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 7 6 8)(9 11)(10 12)(13 14 15 16)(17 20 19 18)
(1 12 15)(2 10 13)(3 9 14)(4 11 16)(5 21 17)(6 23 19)(7 22 18)(8 24 20)
G:=sub<Sym(24)| (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,19)(14,18)(15,17)(16,20), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,12,15)(2,10,13)(3,9,14)(4,11,16)(5,21,17)(6,23,19)(7,22,18)(8,24,20)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,19)(14,18)(15,17)(16,20), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,20,19,18), (1,12,15)(2,10,13)(3,9,14)(4,11,16)(5,21,17)(6,23,19)(7,22,18)(8,24,20) );
G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,12),(10,11),(13,16),(14,15),(17,18),(19,20),(21,22),(23,24)], [(1,5),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,21),(13,19),(14,18),(15,17),(16,20)], [(1,2),(3,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,20,19,18)], [(1,12,15),(2,10,13),(3,9,14),(4,11,16),(5,21,17),(6,23,19),(7,22,18),(8,24,20)])
G:=TransitiveGroup(24,380);
►On 24 points - transitive group 24T382
Generators in S24
(1 5)(2 6)(3 4)(7 8)(9 20)(10 19)(11 18)(12 17)(13 22)(14 21)(15 24)(16 23)
(1 7)(2 4)(3 6)(5 8)(9 19)(10 20)(11 17)(12 18)(13 16)(14 15)(21 24)(22 23)
(1 6)(2 5)(3 7)(4 8)(13 15)(14 16)(21 23)(22 24)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(17 19)(18 20)
(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 3 6 7)(2 4 5 8)(9 20)(10 17)(11 18)(12 19)(13 21 15 23)(14 22 16 24)
(1 11 15)(2 20 22)(3 19 16)(4 10 23)(5 18 24)(6 9 13)(7 17 14)(8 12 21)
G:=sub<Sym(24)| (1,5)(2,6)(3,4)(7,8)(9,20)(10,19)(11,18)(12,17)(13,22)(14,21)(15,24)(16,23), (1,7)(2,4)(3,6)(5,8)(9,19)(10,20)(11,17)(12,18)(13,16)(14,15)(21,24)(22,23), (1,6)(2,5)(3,7)(4,8)(13,15)(14,16)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,6,7)(2,4,5,8)(9,20)(10,17)(11,18)(12,19)(13,21,15,23)(14,22,16,24), (1,11,15)(2,20,22)(3,19,16)(4,10,23)(5,18,24)(6,9,13)(7,17,14)(8,12,21)>;
G:=Group( (1,5)(2,6)(3,4)(7,8)(9,20)(10,19)(11,18)(12,17)(13,22)(14,21)(15,24)(16,23), (1,7)(2,4)(3,6)(5,8)(9,19)(10,20)(11,17)(12,18)(13,16)(14,15)(21,24)(22,23), (1,6)(2,5)(3,7)(4,8)(13,15)(14,16)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,6,7)(2,4,5,8)(9,20)(10,17)(11,18)(12,19)(13,21,15,23)(14,22,16,24), (1,11,15)(2,20,22)(3,19,16)(4,10,23)(5,18,24)(6,9,13)(7,17,14)(8,12,21) );
G=PermutationGroup([(1,5),(2,6),(3,4),(7,8),(9,20),(10,19),(11,18),(12,17),(13,22),(14,21),(15,24),(16,23)], [(1,7),(2,4),(3,6),(5,8),(9,19),(10,20),(11,17),(12,18),(13,16),(14,15),(21,24),(22,23)], [(1,6),(2,5),(3,7),(4,8),(13,15),(14,16),(21,23),(22,24)], [(1,6),(2,5),(3,7),(4,8),(9,11),(10,12),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,6,7),(2,4,5,8),(9,20),(10,17),(11,18),(12,19),(13,21,15,23),(14,22,16,24)], [(1,11,15),(2,20,22),(3,19,16),(4,10,23),(5,18,24),(6,9,13),(7,17,14),(8,12,21)])
G:=TransitiveGroup(24,382);
Matrix representation ►G ⊆ GL12(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0],[0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0],[0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0] >;
Character table of C24.6A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 3 | 4 | 4 | 4 | 12 | 16 | 16 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | 1 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | 1 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ13 | 3 | 3 | -3 | -3 | 3 | 1 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | -3 | 3 | -3 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | 3 | -3 | -3 | 1 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ16 | 3 | 3 | 3 | 3 | 3 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ17 | 12 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
In GAP, Magma, Sage, TeX
C_2^4._6A_4
% in TeX
G:=Group("C2^4.6A4"); // GroupNames label
G:=SmallGroup(192,1008);
// by ID
G=gap.SmallGroup(192,1008);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,850,185,360,2524,2111,1173,102,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=d*c=g*c*g^-1=c*d,f^2=g*d*g^-1=c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*g=g*a,e*b*e^-1=b*c=c*b,f*b*f^-1=b*d=d*b,b*g=g*b,c*e=e*c,c*f=f*c,g*f*g^-1=d*e=e*d,d*f=f*d,e*f=f*e,g*e*g^-1=c*e*f>;
// generators/relations
×
⋊
ℤ
𝔽
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≀