Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#518 ⋊ C2 |
| Sylow | C3 ⋊ C4×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting | C3×#518 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C2×#1004 | C2 |
| hypercenter | #518 | D3 |
| hypocenter | C3 | C4×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #518 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(2,3)(4,6,7,5);(1,2,3)(9,13)(12,14);(10,15)(12,14);(8,10)(9,12)(11,15)(13,14);(4,5,7,6)(8,9)(10,14)(11,13)(12,15)Polycyclic presentation:
2^7,3|a^2=g,e^2=g,b^a=bf,d^a=df,c^b=cf,h^e=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10801060, #10801066, #10801071, … |
| maximal normal subgroup of | #10801060, #10801066, #10801071, … |
| maximal quotient of | #10953417, #10953421, Dic3×#518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18433);