Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#424 ⋊ C2 |
| Sylow | C3 ⋊ #2929 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#424 | C2 |
| prime radical | #5524 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #424 | D3 |
| hypocenter | C3 | #2929 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
520 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 185 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 — 4 generators on 23 points:
(1,2,3)(9,14)(13,19)(15,21)(20,23);(2,3)(8,10,11,16,12,17,18,22)(9,13,15,20,14,19,21,23);(2,3)(5,7);(4,5)(6,7)(8,9,11,15,12,14,18,21)(10,13,16,20,17,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^b=h^2,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10704423, #10704427, #10704431, … |
| maximal normal subgroup of | #10704423, #10704427, #10704431, … |
| maximal quotient of | #10765054, #10765063, #10765072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11649);