Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#398 ⋊ C2 |
| Sylow | C3 ⋊ #2587 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#398 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #398 | D3 |
| hypocenter | C3 | #2587 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #398 | |
428 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)(5,9)(8,10)(13,16)(17,19)(20,22,23,27)(21,25,24,26);(12,14,15,18)(13,17,16,19)(21,24)(25,26);(4,6,7,11)(5,8,9,10)(13,16)(14,18)(21,24)(25,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|b^2=ef,c^2=g,d^2=fg,b^a=be,c^a=cfg,d^a=de,h^a=h^2,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2587, C2×#77802, #10965535, … |
| maximal normal subgroup of | D3×#2587, C2×#77802, #10965535, … |
| maximal quotient of | #10717656, #10717805, #10719016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10378);