Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#423 ⋊ C2 |
| Sylow | C3 ⋊ #2905 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#423 | C2 |
| prime radical | C23×D3 | C2×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #423 | D3 |
| hypocenter | C3 | #2905 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #423 | |
484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 484 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) = 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)(10,16)(13,19)(17,22)(20,23);(4,5,6,7)(8,10)(9,13)(11,17)(12,16)(14,19)(15,20)(18,22)(21,23);(5,7)(8,9,11,15,12,14,18,21)(10,13,17,20,16,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,f^2=g,b^a=be,d^a=dg,c^b=cg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10689205, #10689208, #10689210, … |
| maximal normal subgroup of | #10689205, #10689208, #10689210, … |
| maximal quotient of | #10764330, #10764334, #10764338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11624);