Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#396 ⋊ C2 |
| Sylow | C3 ⋊ #2837 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2837 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(12,14,15,18)(13,17,16,19)(21,25)(24,27);(6,10)(8,11)(14,18)(17,19)(20,22,23,26)(21,24,25,27);(2,3)(4,6)(5,8)(7,10)(9,11)(13,16)(14,18)(20,21)(22,24)(23,25)(26,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,16)(14,17,18,19)(22,26)(24,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=eg,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,d^b=defg,h^b=h^2,d^c=deg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2837, C2×#77409, #11001480, … |
| maximal normal subgroup of | D3×#2837, C2×#77409, #11001480, … |
| maximal quotient of | #10725906, #10725919, #10725944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9985);