Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C43 ⋊ C2 |
| Sylow | C3 ⋊ #2866 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C43 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #2866 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C43 | |
418 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)(13,17)(16,18)(20,22,23,26)(21,25,24,27);(12,14,15,19)(13,16,17,18)(21,24)(25,27);(4,6,7,10)(5,8,9,11)(21,24)(25,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,17)(16,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|b^2=ef,c^2=efg,d^2=e,b^a=be,c^a=cfg,d^a=dg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10968524, #10968526, #10968557, … |
| maximal normal subgroup of | #10968524, #10968526, #10968557, … |
| maximal quotient of | #10726391, #10726468, #10726499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5893);