Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2720 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2720 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
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:
(1,2,3)(8,11,12,10)(9,14,13,15)(17,22)(21,23);(2,3)(5,7)(8,10,12,11)(9,14,13,15)(16,19,20,18)(17,21,22,23);(4,5)(6,7)(9,13)(10,11)(16,18,20,19)(17,21,22,23);(8,9,12,13)(10,15,11,14)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002617, #11002630, #11002655, … |
| maximal normal subgroup of | #11002617, #11002630, #11002655, … |
| maximal quotient of | #10722880, #10723062, #10723123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15379);