Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#446 ⋊ C2 |
| Sylow | C3 ⋊ #2965 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#446 | C2 |
| prime radical | C2×#239 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #446 | D3 |
| hypocenter | C3 | #2965 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #446 | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(4,6,7,10)(5,9,8,11);(12,14,15,17,16,18,19,13);(2,3)(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,15,14,16,17,19,18)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cg,h^a=h^2,h^b=h^2,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10705038, #10705048, #10705083, … |
| maximal normal subgroup of | #10705038, #10705048, #10705083, … |
| maximal quotient of | #10793396, #10793405, #10793531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12514);