Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2724 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#390 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | #2724 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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)(10,15)(12,14)(17,22)(21,23);(2,3)(5,7)(9,13)(10,15)(16,19,20,18)(17,21,22,23);(2,3)(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,18,20,19)(17,21,22,23);(8,9)(10,14)(11,13)(12,15)(16,17,20,22)(18,21,19,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=e,b^a=bef,c^a=cf,d^a=de,c^b=cg,d^b=def,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10683894, #10683897, #10683977, … |
| maximal normal subgroup of | #10683894, #10683897, #10683977, … |
| maximal quotient of | #10722854, #10723604, #10723656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8974);