Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2735 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2735 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
552 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(5,9)(6,11)(13,17)(16,19);(13,17)(16,19)(22,26)(24,27);(2,3)(4,6)(5,8)(7,11)(9,10)(12,14,15,18)(13,16,17,19)(20,22)(21,24)(23,26)(25,27);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,24,26,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707834, #10707838, #10708318, … |
| maximal normal subgroup of | #10707834, #10707838, #10708318, … |
| maximal quotient of | #10722584, #10722590, #10724034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10696);