Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#389 ⋊ C2 |
| Sylow | C3 ⋊ #2807 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#389 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #389 | D3 |
| hypocenter | C3 | #2807 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #389 | |
408 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)(6,11)(8,10)(20,23,24,22)(21,26,27,25);(5,9)(8,10)(12,14,15,18)(13,16,17,19)(20,22,24,23)(21,25,27,26);(2,3)(4,6)(5,8)(7,11)(9,10)(14,18)(16,19);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,17)(14,16,18,19)(20,21,24,27)(22,26,23,25)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=eg,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 | #11002801, #11002810, #11002833, … |
| maximal normal subgroup of | #11002801, #11002810, #11002833, … |
| maximal quotient of | #10723619, #10723625, #10723772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8858);