Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #2678 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#121 | C2 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #2678 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(5,7)(13,17)(14,19);(2,3)(5,7)(9,11);(8,9)(10,11)(12,14)(13,16)(15,19)(17,18);(2,3)(4,5,6,7)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|a^2=f,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11001924, #11001930, #11002076, … |
| maximal normal subgroup of | #11001924, #11001930, #11002076, … |
| maximal quotient of | #10722475, #10722478, #10722489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15296);