Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C22×C3×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #3492 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C22×C3×C4⋊C4 | C2 |
| hypercenter | C22×C4⋊C4 | D3 |
| hypocenter | C3 | #3492 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(13,16)(17,19);(12,14,15,18)(13,17,16,19);(5,9)(8,11);(4,6,7,10)(5,8,9,11)(13,16)(14,18);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|b^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10686702, #10686705, #10687318, … |
| maximal normal subgroup of | #10686702, #10686705, #10687318, … |
| maximal quotient of | #10955248, #10955469, #10955681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18294);