Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#388 ⋊ C2 |
| Sylow | C3 ⋊ #2413 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#388 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2413 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
610 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)(13,18)(17,19);(5,7)(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)(12,13)(14,19)(15,17)(16,18)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^b=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2413, C2×#75843, #10963897, … |
| maximal normal subgroup of | D3×#2413, C2×#75843, #10963897, … |
| maximal quotient of | #10717076, #10717921, #10718011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8419);