Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Fitting | C3×#322 ⋊ C2 |
| Sylow | C3 ⋊ #1455 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D3 |
| commutator, socle | C22×C3 | C2×C42 |
| Fitting | C3×#322 | C2 |
| prime radical | C22×D3 | C42 |
| cosocle | C3×C42 | C23 |
| hypercenter | #322 | D3 |
| hypocenter | C3 | #1455 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #322 | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(5,8)(9,11)(13,16)(17,19);(12,13,14,17,15,16,18,19);(2,3)(4,5,6,9,7,8,10,11)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cf,h^a=h^2,c^b=cg,h^c=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | #10705266, D3×#1455, C2×#68075, … |
| maximal normal subgroup of | #10705266, D3×#1455, C2×#68075, … |
| maximal quotient of | #10684564, #10684579, #10685623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 651);