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 ⋊ #3500 |
| 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 | #3500 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(8,11,12,20)(9,15,14,19)(10,18,17,23)(13,21,16,22);(10,17)(13,16)(18,23)(21,22);(2,3)(4,5,6,7)(9,14)(13,16)(15,19)(21,22);(8,10)(9,13)(11,18)(12,17)(14,16)(15,21)(19,22)(20,23);(5,7)(8,9)(10,16)(11,19)(12,14)(13,17)(15,20)(18,21)(22,23)Polycyclic presentation:
2^7,3|c^2=fg,e^2=f,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 | #10710757, #10804389, #10804396, … |
| maximal normal subgroup of | #10710757, #10804389, #10804396, … |
| maximal quotient of | #10955294, #10955569, #10955602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19859);