direct product, metabelian, soluble, monomial
Aliases: C2×C23.A4, C24.5A4, C4⋊1D4⋊2C6, (C2×C42)⋊2C6, C42⋊3(C2×C6), C42⋊C3⋊5C22, C23.4(C2×A4), C22.4(C22×A4), (C2×C4⋊1D4)⋊C3, (C2×C42⋊C3)⋊2C2, SmallGroup(192,1002)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C2×C23.A4 |
Subgroups: 546 in 108 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22, C22 [×16], C6 [×3], C2×C4 [×6], D4 [×16], C23, C23 [×2], C23 [×10], A4, C2×C6, C42, C42, C22×C4, C2×D4 [×16], C24, C24, C2×A4 [×3], C2×C42, C4⋊1D4 [×2], C4⋊1D4 [×2], C22×D4 [×2], C42⋊C3, C22×A4, C2×C4⋊1D4, C2×C42⋊C3, C23.A4 [×2], C2×C23.A4
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C23.A4, C2×C23.A4
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, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, bd=db, ebe-1=bcd, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >
►On 12 points - transitive group 12T89
Generators in S12
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 3)(2 4)(5 8)(6 7)(9 12)(10 11)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3 2 4)(9 12 11 10)
(1 8 12)(2 6 10)(3 5 9)(4 7 11)
G:=sub<Sym(12)| (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(5,8)(6,7)(9,12)(10,11), (1,2)(3,4)(9,11)(10,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3,2,4)(9,12,11,10), (1,8,12)(2,6,10)(3,5,9)(4,7,11)>;
G:=Group( (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(5,8)(6,7)(9,12)(10,11), (1,2)(3,4)(9,11)(10,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3,2,4)(9,12,11,10), (1,8,12)(2,6,10)(3,5,9)(4,7,11) );
G=PermutationGroup([(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)], [(1,3),(2,4),(5,8),(6,7),(9,12),(10,11)], [(1,2),(3,4),(9,11),(10,12)], [(1,2),(3,4),(5,7),(6,8)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3,2,4),(9,12,11,10)], [(1,8,12),(2,6,10),(3,5,9),(4,7,11)])
G:=TransitiveGroup(12,89);
►On 12 points - transitive group 12T92
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(2 4)(5 7)(10 12)
(1 3)(2 4)(9 11)(10 12)
(1 3)(2 4)(5 7)(6 8)
(5 6 7 8)(9 10 11 12)
(1 2 3 4)(5 7)(6 8)(9 10 11 12)
(1 8 11)(2 7 12)(3 6 9)(4 5 10)
G:=sub<Sym(12)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (2,4)(5,7)(10,12), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,8,11)(2,7,12)(3,6,9)(4,5,10)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (2,4)(5,7)(10,12), (1,3)(2,4)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,8,11)(2,7,12)(3,6,9)(4,5,10) );
G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(2,4),(5,7),(10,12)], [(1,3),(2,4),(9,11),(10,12)], [(1,3),(2,4),(5,7),(6,8)], [(5,6,7,8),(9,10,11,12)], [(1,2,3,4),(5,7),(6,8),(9,10,11,12)], [(1,8,11),(2,7,12),(3,6,9),(4,5,10)])
G:=TransitiveGroup(12,92);
►On 24 points - transitive group 24T453
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 6)(2 5)(3 7)(4 8)(9 21)(10 24)(11 23)(12 22)(13 17)(14 20)(15 19)(16 18)
(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)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 2 4)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 23 15)(2 21 13)(3 22 16)(4 24 14)(5 9 17)(6 11 19)(7 12 18)(8 10 20)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18), (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), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,2,4)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,23,15)(2,21,13)(3,22,16)(4,24,14)(5,9,17)(6,11,19)(7,12,18)(8,10,20)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,6)(2,5)(3,7)(4,8)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18), (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), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,2,4)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,23,15)(2,21,13)(3,22,16)(4,24,14)(5,9,17)(6,11,19)(7,12,18)(8,10,20) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,6),(2,5),(3,7),(4,8),(9,21),(10,24),(11,23),(12,22),(13,17),(14,20),(15,19),(16,18)], [(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)], [(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,2,4),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,23),(22,24)], [(1,23,15),(2,21,13),(3,22,16),(4,24,14),(5,9,17),(6,11,19),(7,12,18),(8,10,20)])
G:=TransitiveGroup(24,453);
►On 24 points - transitive group 24T454
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 4)(2 3)(5 7)(6 8)(9 22)(10 21)(11 24)(12 23)(13 16)(14 15)(17 20)(18 19)
(1 7)(2 8)(3 6)(4 5)(13 15)(14 16)(17 19)(18 20)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(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 5 7 4)(2 6 8 3)(9 21)(10 22)(11 23)(12 24)(13 20 15 18)(14 17 16 19)
(1 10 19)(2 24 13)(3 11 16)(4 21 18)(5 23 20)(6 9 14)(7 12 17)(8 22 15)
G:=sub<Sym(24)| (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,4)(2,3)(5,7)(6,8)(9,22)(10,21)(11,24)(12,23)(13,16)(14,15)(17,20)(18,19), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(17,19)(18,20), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(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), (1,5,7,4)(2,6,8,3)(9,21)(10,22)(11,23)(12,24)(13,20,15,18)(14,17,16,19), (1,10,19)(2,24,13)(3,11,16)(4,21,18)(5,23,20)(6,9,14)(7,12,17)(8,22,15)>;
G:=Group( (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,4)(2,3)(5,7)(6,8)(9,22)(10,21)(11,24)(12,23)(13,16)(14,15)(17,20)(18,19), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(17,19)(18,20), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(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), (1,5,7,4)(2,6,8,3)(9,21)(10,22)(11,23)(12,24)(13,20,15,18)(14,17,16,19), (1,10,19)(2,24,13)(3,11,16)(4,21,18)(5,23,20)(6,9,14)(7,12,17)(8,22,15) );
G=PermutationGroup([(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,4),(2,3),(5,7),(6,8),(9,22),(10,21),(11,24),(12,23),(13,16),(14,15),(17,20),(18,19)], [(1,7),(2,8),(3,6),(4,5),(13,15),(14,16),(17,19),(18,20)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(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)], [(1,5,7,4),(2,6,8,3),(9,21),(10,22),(11,23),(12,24),(13,20,15,18),(14,17,16,19)], [(1,10,19),(2,24,13),(3,11,16),(4,21,18),(5,23,20),(6,9,14),(7,12,17),(8,22,15)])
G:=TransitiveGroup(24,454);
►On 24 points - transitive group 24T455
Generators in S24
(1 8)(2 6)(3 7)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 6)(2 8)(3 5)(4 7)(9 22)(10 21)(11 24)(12 23)(13 17)(14 20)(15 19)(16 18)
(1 3)(2 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 3 2)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 24 20)(2 21 19)(3 22 18)(4 23 17)(5 9 16)(6 11 14)(7 12 13)(8 10 15)
G:=sub<Sym(24)| (1,8)(2,6)(3,7)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,6)(2,8)(3,5)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18), (1,3)(2,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,3,2)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,24,20)(2,21,19)(3,22,18)(4,23,17)(5,9,16)(6,11,14)(7,12,13)(8,10,15)>;
G:=Group( (1,8)(2,6)(3,7)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,6)(2,8)(3,5)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18), (1,3)(2,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,3,2)(5,7,6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,24,20)(2,21,19)(3,22,18)(4,23,17)(5,9,16)(6,11,14)(7,12,13)(8,10,15) );
G=PermutationGroup([(1,8),(2,6),(3,7),(4,5),(9,23),(10,24),(11,21),(12,22),(13,18),(14,19),(15,20),(16,17)], [(1,6),(2,8),(3,5),(4,7),(9,22),(10,21),(11,24),(12,23),(13,17),(14,20),(15,19),(16,18)], [(1,3),(2,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,3,2),(5,7,6,8),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,23),(22,24)], [(1,24,20),(2,21,19),(3,22,18),(4,23,17),(5,9,16),(6,11,14),(7,12,13),(8,10,15)])
G:=TransitiveGroup(24,455);
►On 24 points - transitive group 24T456
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 21)(10 22)(11 23)(12 24)(13 19)(14 20)(15 17)(16 18)
(1 5)(2 6)(3 8)(4 7)(9 22)(10 21)(11 24)(12 23)(13 16)(14 15)(17 20)(18 19)
(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(21 23)(22 24)
(1 8)(2 7)(3 5)(4 6)(13 15)(14 16)(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 6 8 4)(2 5 7 3)(9 22 11 24)(10 23 12 21)(13 17)(14 18)(15 19)(16 20)
(1 13 11)(2 19 23)(3 14 22)(4 20 10)(5 16 24)(6 18 12)(7 17 21)(8 15 9)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,5)(2,6)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,16)(14,15)(17,20)(18,19), (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(21,23)(22,24), (1,8)(2,7)(3,5)(4,6)(13,15)(14,16)(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), (1,6,8,4)(2,5,7,3)(9,22,11,24)(10,23,12,21)(13,17)(14,18)(15,19)(16,20), (1,13,11)(2,19,23)(3,14,22)(4,20,10)(5,16,24)(6,18,12)(7,17,21)(8,15,9)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,5)(2,6)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,16)(14,15)(17,20)(18,19), (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(21,23)(22,24), (1,8)(2,7)(3,5)(4,6)(13,15)(14,16)(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), (1,6,8,4)(2,5,7,3)(9,22,11,24)(10,23,12,21)(13,17)(14,18)(15,19)(16,20), (1,13,11)(2,19,23)(3,14,22)(4,20,10)(5,16,24)(6,18,12)(7,17,21)(8,15,9) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,21),(10,22),(11,23),(12,24),(13,19),(14,20),(15,17),(16,18)], [(1,5),(2,6),(3,8),(4,7),(9,22),(10,21),(11,24),(12,23),(13,16),(14,15),(17,20),(18,19)], [(1,8),(2,7),(3,5),(4,6),(9,11),(10,12),(21,23),(22,24)], [(1,8),(2,7),(3,5),(4,6),(13,15),(14,16),(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)], [(1,6,8,4),(2,5,7,3),(9,22,11,24),(10,23,12,21),(13,17),(14,18),(15,19),(16,20)], [(1,13,11),(2,19,23),(3,14,22),(4,20,10),(5,16,24),(6,18,12),(7,17,21),(8,15,9)])
G:=TransitiveGroup(24,456);
►On 24 points - transitive group 24T463
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(3 5)(4 6)(10 12)(13 15)(18 20)(22 24)
(1 8)(2 7)(3 5)(4 6)(17 19)(18 20)(21 23)(22 24)
(1 8)(2 7)(3 5)(4 6)(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 4 8 6)(2 3 7 5)(9 16)(10 13)(11 14)(12 15)(17 22 19 24)(18 23 20 21)
(1 11 17)(2 16 21)(3 10 22)(4 15 18)(5 12 24)(6 13 20)(7 14 23)(8 9 19)
G:=sub<Sym(24)| (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (3,5)(4,6)(10,12)(13,15)(18,20)(22,24), (1,8)(2,7)(3,5)(4,6)(17,19)(18,20)(21,23)(22,24), (1,8)(2,7)(3,5)(4,6)(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), (1,4,8,6)(2,3,7,5)(9,16)(10,13)(11,14)(12,15)(17,22,19,24)(18,23,20,21), (1,11,17)(2,16,21)(3,10,22)(4,15,18)(5,12,24)(6,13,20)(7,14,23)(8,9,19)>;
G:=Group( (1,8)(2,7)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (3,5)(4,6)(10,12)(13,15)(18,20)(22,24), (1,8)(2,7)(3,5)(4,6)(17,19)(18,20)(21,23)(22,24), (1,8)(2,7)(3,5)(4,6)(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), (1,4,8,6)(2,3,7,5)(9,16)(10,13)(11,14)(12,15)(17,22,19,24)(18,23,20,21), (1,11,17)(2,16,21)(3,10,22)(4,15,18)(5,12,24)(6,13,20)(7,14,23)(8,9,19) );
G=PermutationGroup([(1,8),(2,7),(3,5),(4,6),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(3,5),(4,6),(10,12),(13,15),(18,20),(22,24)], [(1,8),(2,7),(3,5),(4,6),(17,19),(18,20),(21,23),(22,24)], [(1,8),(2,7),(3,5),(4,6),(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)], [(1,4,8,6),(2,3,7,5),(9,16),(10,13),(11,14),(12,15),(17,22,19,24),(18,23,20,21)], [(1,11,17),(2,16,21),(3,10,22),(4,15,18),(5,12,24),(6,13,20),(7,14,23),(8,9,19)])
G:=TransitiveGroup(24,463);
►On 24 points - transitive group 24T464
Generators in S24
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 7)(2 8)(3 4)(5 6)(9 14)(10 13)(11 16)(12 15)(17 24)(18 23)(19 22)(20 21)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 8)(2 7)(3 6)(4 5)(17 19)(18 20)(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 5 8 4)(2 6 7 3)(9 15 11 13)(10 16 12 14)(17 24)(18 21)(19 22)(20 23)
(1 24 13)(2 19 12)(3 21 11)(4 20 16)(5 18 14)(6 23 9)(7 17 10)(8 22 15)
G:=sub<Sym(24)| (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,7)(2,8)(3,4)(5,6)(9,14)(10,13)(11,16)(12,15)(17,24)(18,23)(19,22)(20,21), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,8)(2,7)(3,6)(4,5)(17,19)(18,20)(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), (1,5,8,4)(2,6,7,3)(9,15,11,13)(10,16,12,14)(17,24)(18,21)(19,22)(20,23), (1,24,13)(2,19,12)(3,21,11)(4,20,16)(5,18,14)(6,23,9)(7,17,10)(8,22,15)>;
G:=Group( (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,7)(2,8)(3,4)(5,6)(9,14)(10,13)(11,16)(12,15)(17,24)(18,23)(19,22)(20,21), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,8)(2,7)(3,6)(4,5)(17,19)(18,20)(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), (1,5,8,4)(2,6,7,3)(9,15,11,13)(10,16,12,14)(17,24)(18,21)(19,22)(20,23), (1,24,13)(2,19,12)(3,21,11)(4,20,16)(5,18,14)(6,23,9)(7,17,10)(8,22,15) );
G=PermutationGroup([(1,8),(2,7),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,7),(2,8),(3,4),(5,6),(9,14),(10,13),(11,16),(12,15),(17,24),(18,23),(19,22),(20,21)], [(1,8),(2,7),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16)], [(1,8),(2,7),(3,6),(4,5),(17,19),(18,20),(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)], [(1,5,8,4),(2,6,7,3),(9,15,11,13),(10,16,12,14),(17,24),(18,21),(19,22),(20,23)], [(1,24,13),(2,19,12),(3,21,11),(4,20,16),(5,18,14),(6,23,9),(7,17,10),(8,22,15)])
G:=TransitiveGroup(24,464);
►On 24 points - transitive group 24T465
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(1 8)(2 7)(3 4)(5 6)(9 24)(10 23)(11 22)(12 21)(13 19)(14 18)(15 17)(16 20)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(21 23)(22 24)
(1 7)(2 8)(3 6)(4 5)(13 15)(14 16)(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 7 6)(2 4 8 5)(9 23 11 21)(10 24 12 22)(13 17)(14 18)(15 19)(16 20)
(1 16 9)(2 18 22)(3 17 10)(4 15 23)(5 13 21)(6 19 12)(7 14 11)(8 20 24)
G:=sub<Sym(24)| (1,8)(2,7)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (1,8)(2,7)(3,4)(5,6)(9,24)(10,23)(11,22)(12,21)(13,19)(14,18)(15,17)(16,20), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(21,23)(22,24), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(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), (1,3,7,6)(2,4,8,5)(9,23,11,21)(10,24,12,22)(13,17)(14,18)(15,19)(16,20), (1,16,9)(2,18,22)(3,17,10)(4,15,23)(5,13,21)(6,19,12)(7,14,11)(8,20,24)>;
G:=Group( (1,8)(2,7)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (1,8)(2,7)(3,4)(5,6)(9,24)(10,23)(11,22)(12,21)(13,19)(14,18)(15,17)(16,20), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(21,23)(22,24), (1,7)(2,8)(3,6)(4,5)(13,15)(14,16)(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), (1,3,7,6)(2,4,8,5)(9,23,11,21)(10,24,12,22)(13,17)(14,18)(15,19)(16,20), (1,16,9)(2,18,22)(3,17,10)(4,15,23)(5,13,21)(6,19,12)(7,14,11)(8,20,24) );
G=PermutationGroup([(1,8),(2,7),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,17),(14,18),(15,19),(16,20)], [(1,8),(2,7),(3,4),(5,6),(9,24),(10,23),(11,22),(12,21),(13,19),(14,18),(15,17),(16,20)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(21,23),(22,24)], [(1,7),(2,8),(3,6),(4,5),(13,15),(14,16),(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)], [(1,3,7,6),(2,4,8,5),(9,23,11,21),(10,24,12,22),(13,17),(14,18),(15,19),(16,20)], [(1,16,9),(2,18,22),(3,17,10),(4,15,23),(5,13,21),(6,19,12),(7,14,11),(8,20,24)])
G:=TransitiveGroup(24,465);
►On 24 points - transitive group 24T466
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 13)(10 14)(11 15)(12 16)(17 24)(18 21)(19 22)(20 23)
(1 2)(7 8)(9 11)(13 15)(18 20)(21 23)
(1 2)(3 4)(5 6)(7 8)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(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 4 2 3)(5 8 6 7)(17 20 19 18)(21 24 23 22)
(1 11 18)(2 9 20)(3 10 17)(4 12 19)(5 16 22)(6 14 24)(7 15 21)(8 13 23)
G:=sub<Sym(24)| (1,7)(2,8)(3,6)(4,5)(9,13)(10,14)(11,15)(12,16)(17,24)(18,21)(19,22)(20,23), (1,2)(7,8)(9,11)(13,15)(18,20)(21,23), (1,2)(3,4)(5,6)(7,8)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(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), (1,4,2,3)(5,8,6,7)(17,20,19,18)(21,24,23,22), (1,11,18)(2,9,20)(3,10,17)(4,12,19)(5,16,22)(6,14,24)(7,15,21)(8,13,23)>;
G:=Group( (1,7)(2,8)(3,6)(4,5)(9,13)(10,14)(11,15)(12,16)(17,24)(18,21)(19,22)(20,23), (1,2)(7,8)(9,11)(13,15)(18,20)(21,23), (1,2)(3,4)(5,6)(7,8)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(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), (1,4,2,3)(5,8,6,7)(17,20,19,18)(21,24,23,22), (1,11,18)(2,9,20)(3,10,17)(4,12,19)(5,16,22)(6,14,24)(7,15,21)(8,13,23) );
G=PermutationGroup([(1,7),(2,8),(3,6),(4,5),(9,13),(10,14),(11,15),(12,16),(17,24),(18,21),(19,22),(20,23)], [(1,2),(7,8),(9,11),(13,15),(18,20),(21,23)], [(1,2),(3,4),(5,6),(7,8),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(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)], [(1,4,2,3),(5,8,6,7),(17,20,19,18),(21,24,23,22)], [(1,11,18),(2,9,20),(3,10,17),(4,12,19),(5,16,22),(6,14,24),(7,15,21),(8,13,23)])
G:=TransitiveGroup(24,466);
Matrix representation ►G ⊆ GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 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 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 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 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0] >;
Character table of C2×C23.A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 12 | 12 | 16 | 16 | 6 | 6 | 6 | 6 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 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 | 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 | -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 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | linear of order 6 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ10 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | linear of order 6 |
| ρ11 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ13 | 3 | -3 | -3 | 3 | -3 | 3 | 1 | -1 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 3 | -3 | -3 | 3 | 3 | -3 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ16 | 3 | 3 | 3 | 3 | -3 | -3 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
| ρ18 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
| ρ19 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
In GAP, Magma, Sage, TeX
C_2\times C_2^3.A_4
% in TeX
G:=Group("C2xC2^3.A4"); // GroupNames label
G:=SmallGroup(192,1002);
// by ID
G=gap.SmallGroup(192,1002);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,185,360,2524,1173,102,1027,1784]);
// 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,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f^-1=b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,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
○
≀