Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2744 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2744 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
392 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)(5,9)(8,11)(21,24)(25,27);(5,9)(8,11)(14,18)(16,19)(20,22,23,26)(21,25,24,27);(2,3)(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,18)(17,19)(20,21,23,24)(22,25,26,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,16,18,19)(21,24)(22,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=fg,c^2=ef,b^a=bef,c^a=cfg,d^a=deg,h^a=h^2,c^b=cg,d^b=def,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002305, #11002324, #11002407, … |
| maximal normal subgroup of | #11002305, #11002324, #11002407, … |
| maximal quotient of | #10722605, #10722665, #10722811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7121);