Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2344 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×#117 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×#117 | D3 |
| hypocenter | C3 | #2344 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(8,10,11,9);(13,17)(16,19);(9,10)(12,14,15,18)(13,16,17,19);(2,3)(4,5,6,7)(8,9,11,10)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^d=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #10810526, #10810527, #10811334, … |
| maximal normal subgroup of | #10810526, #10810527, #10811334, … |
| maximal quotient of | #10716592, #10716596, C22⋊C4×Dic3⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14814);