Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#380 ⋊ C2 |
| Sylow | C3 ⋊ #2352 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#380 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×#67 | C22 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2352 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(9,11);(12,15,16,14)(13,17,18,19);(2,3)(8,9)(10,11)(12,14,16,15)(13,17,18,19);(2,3)(4,5,6,7)(9,11)(12,13)(14,17)(15,19)(16,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=f,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10810608, #10810610, #10810634, … |
| maximal normal subgroup of | #10810608, #10810610, #10810634, … |
| maximal quotient of | #10716606, #10716611, #10716632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6459);