Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2484 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C22×C3×C4⋊C4 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C22×C4⋊C4 | D3 |
| hypocenter | C3 | #2484 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
512 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 512 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(17,22)(21,23);(5,7)(10,14)(12,15)(16,19,20,18)(17,21,22,23);(8,10)(9,12)(11,14)(13,15)(16,18,20,19)(17,21,22,23);(2,3)(4,5)(6,7)(8,9,11,13)(10,12,14,15)(16,17)(18,23)(19,21)(20,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10692999, #10693001, #10693019, … |
| maximal normal subgroup of | #10692999, #10693001, #10693019, … |
| maximal quotient of | #10717078, #10717888, #10718264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14752);