Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #3484 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×H(5,𝔽2) | C2 |
| hypercenter | C2×H(5,𝔽2) | D3 |
| hypocenter | C3 | #3484 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(9,13)(12,14);(10,15)(12,14);(2,3)(5,7);(8,10)(9,12)(11,15)(13,14);(4,5)(6,7)(8,9)(10,14)(11,13)(12,15)Polycyclic presentation:
2^7,3|b^a=bf,c^a=cg,e^a=ef,d^b=df,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10710732, #10804335, #10804347, … |
| maximal normal subgroup of | #10710732, #10804335, #10804347, … |
| maximal quotient of | #10955196, #10955440, #10955464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19860);