Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2704 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2704 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(5,7)(9,13)(12,15)(17,20)(21,23);(2,3)(4,5)(6,7)(8,10,11,14)(9,12,13,15)(17,20)(18,22);(2,3)(8,9,11,13)(10,12,14,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=ef,b^a=bef,c^a=cf,d^a=de,h^a=h^2,c^b=cg,d^b=def,h^b=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11002169, #11002174, #11002228, … |
| maximal normal subgroup of | #11002169, #11002174, #11002228, … |
| maximal quotient of | #10722531, #10722651, #10722772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9928);