Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2584 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#398 | C2 |
| hypercenter | #398 | D3 |
| hypocenter | C3 | #2584 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #398 | |
468 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 157 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)(8,10,11,15)(9,13,12,14);(2,3)(5,7)(16,18,19,23)(17,20,21,22);(9,12)(13,14)(16,17,19,21)(18,22,23,20);(4,5,6,7)(8,9)(10,14)(11,12)(13,15)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10768931, #10768933, #10768953, … |
| maximal normal subgroup of | #10768931, #10768933, #10768953, … |
| maximal quotient of | #10717592, #10717859, #10718908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10373);