Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1832 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator, socle, prime radical | C23×C3 | C22×C4 |
| Fitting | C3×#376 | C2 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | #376 | D3 |
| hypocenter | C3 | #1832 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,6,7,5)(9,13)(12,15)(18,24)(21,27)(25,30)(28,31);(4,5,7,6)(8,10,11,14)(9,12,13,15)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(2,3)(5,6)(8,9)(10,12)(11,13)(14,15)(16,17,19,22,20,23,26,29)(18,21,25,28,24,27,30,31)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^2,c^b=cg84 conjugacy classes — view the full character table.
| maximal subgroup of | #10689313, #10689314, #10699261, … |
| maximal normal subgroup of | #10689313, #10689314, #10699261, … |
| maximal quotient of | #10688841, #10688845, #10688858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2013);