Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C23×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ #2403 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C23×C3×D4 | C2 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | #2403 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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)(5,7);(9,11)(13,15);(5,7)(12,13)(14,15);(2,3)(4,5)(6,7)(8,9)(10,11)Polycyclic presentation:
2^7,3|b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10693083, #10693084, #10693086, … |
| maximal normal subgroup of | #10693083, #10693084, #10693086, … |
| maximal quotient of | #10717053, #10717057, #10717953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16628);