Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C23×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3486 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C23×C3×Q8 | C2 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #3486 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(9,15)(13,16)(14,23)(21,22);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22);(5,7);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21);(2,3)(4,5)(6,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|b^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687323, #10687325, #10710729, … |
| maximal normal subgroup of | #10687323, #10687325, #10710729, … |
| maximal quotient of | #10955212, #10955469, #10956125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19827);