Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2295 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator | C22×C3 | C23×C4 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×C43 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #2295 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,18,19,22)(17,21,20,23);(8,10,11,15)(9,12,13,14);(4,6,7,5)(17,20)(21,23);(2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=fg,d^2=e,b^a=be,c^a=cfg,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10683132, #10683135, #10683192, … |
| maximal normal subgroup of | #10683132, #10683135, #10683192, … |
| maximal quotient of | #10716298, #10716318, C4×#4933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5876);