Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#123 ⋊ C2 |
| Sylow | C3 ⋊ #2406 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#123 | C2 |
| prime radical | C22×#237 | C2 |
| hypercenter | C2×#123 | D3 |
| hypocenter | C3 | #2406 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
864 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)(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);(4,5)(6,7)(12,13)(14,19)(15,17)(16,18)Polycyclic presentation:
2^7,3|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10712668, #10712688, #10832554, … |
| maximal normal subgroup of | #10712668, #10712688, #10832554, … |
| maximal quotient of | #10717075, #10717875, #10717965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15417);