Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #3481 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4×D4 | C2 |
| hypercenter | C2×C4×D4 | D3 |
| hypocenter | C3 | #3481 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×D4 | |
1242 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)(12,14,15,18)(13,17,16,19);(6,11)(8,10);(5,9)(8,10)(13,16)(17,19);(4,6)(5,8)(7,11)(9,10);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|e^2=fg,b^a=bf,c^a=cg,h^a=h^2,d^b=df90 conjugacy classes — view the full character table.
| maximal subgroup of | #10762153, #10762159, #10762189, … |
| maximal normal subgroup of | #10762153, #10762159, #10762189, … |
| maximal quotient of | #10954629, #10954633, #10954638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18361);