Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C22×C3×C42 ⋊ C2 |
| Sylow | C3 ⋊ #2856 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C22×C3×C42 | C2 |
| hypercenter | C22×C42 | D3 |
| hypocenter | C3 | #2856 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7);(12,13,14,15);(8,9,10,11);(2,3)(4,5)(6,7)(9,11)(13,15)Polycyclic presentation:
2^7,3|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=dg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10683563, #10683565, #10683655, … |
| maximal normal subgroup of | #10683563, #10683565, #10683655, … |
| maximal quotient of | #10726375, #10726439, #10726457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14657);