Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2783 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2783 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,14,15,19)(13,17,16,18)(21,24)(25,27);(2,3)(6,10)(8,11)(13,16)(17,18)(20,22,23,26)(21,25,24,27);(4,6)(5,8)(7,10)(9,11)(20,21,23,24)(22,25,26,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,16)(14,18,19,17)(21,24)(22,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=ef,d^2=eg,b^a=bef,c^a=cfg,d^a=deg,h^a=h^2,c^b=cg,d^b=def,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11003107, #11003113, #11003242, … |
| maximal normal subgroup of | #11003107, #11003113, #11003242, … |
| maximal quotient of | #10723008, #10723083, #10723636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9967);