Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2713 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#125 | C2 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | #2713 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,14)(16,19,20,21)(17,18,22,23);(5,7)(8,10,11,15)(9,12,13,14);(2,3)(4,5)(6,7)(16,18,20,23)(17,21,22,19);(8,9)(10,14)(11,13)(12,15)(16,17,20,22)(18,19,23,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002645, #11002648, #11002666, … |
| maximal normal subgroup of | #11002645, #11002648, #11002666, … |
| maximal quotient of | #10722762, #10723140, #10723168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15624);