Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#99 ⋊ C2 |
| Sylow | C3 ⋊ #3026 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#882 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C2×C3×#99 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | C2×#99 | D3 |
| hypocenter | C3 | #3026 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#99 | |
800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,11)(9,16)(10,17)(15,18);(4,7)(5,10)(6,12)(8,14)(9,15)(11,17)(13,19)(16,18);(2,3)(4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,19)(17,18);(2,3)(4,5)(6,9)(7,10)(8,11)(12,18)(13,16)(14,17)(15,19)Polycyclic presentation:
2^7,3|b^a=be,c^a=cf,d^a=dg,h^a=h^2,f^b=fg,h^b=h^2,e^c=eg51 conjugacy classes — view the full character table.
| maximal subgroup of | #10826735, #10826738, #10826739, … |
| maximal normal subgroup of | #10826735, #10826738, #10826739, … |
| maximal quotient of | #10769023, #10769024, #10769030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11262);