Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C2×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ #2847 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#120 | C2 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2847 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(9,11);(5,7);(2,3)(12,13,14,15);(4,5)(6,7)(8,9)(10,11)(13,15)Polycyclic presentation:
2^7,3|b^2=e,b^a=be,c^a=cf,d^a=dg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10829369, #10829373, #10830817, … |
| maximal normal subgroup of | #10829369, #10829373, #10830817, … |
| maximal quotient of | #10726345, #10726400, #10726403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15177);