Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2570 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2570 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(20,22,23,27)(21,25,24,26);(6,10)(8,11)(12,14,15,19)(13,16,17,18);(2,3)(4,6,7,10)(5,8,9,11)(21,24)(25,26);(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,18,19,16)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=fg,d^2=e,b^a=be,c^a=cfg,d^a=de,c^b=cg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10854229, #10854233, #10854244, … |
| maximal normal subgroup of | #10854229, #10854233, #10854244, … |
| maximal quotient of | #10717452, #10717868, #10718109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6917);