Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#396 ⋊ C2 |
| Sylow | C3 ⋊ #2626 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2626 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(10,14)(13,15);(5,7)(8,10)(9,13)(11,14)(12,15)(16,18,19,23)(17,20,21,22);(4,5)(6,7)(9,12)(13,15);(2,3)(8,9)(10,13)(11,12)(14,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^c=de,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2626, C2×#77424, #10961604, … |
| maximal normal subgroup of | D3×#2626, C2×#77424, #10961604, … |
| maximal quotient of | #10721597, #10721618, #10721620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10000);