Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×#123 ⋊ C2 |
| Sylow | C3 ⋊ #2501 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#123 | C2 |
| prime radical | C2×#939 | C2 |
| hypercenter | C2×#123 | D3 |
| hypocenter | C3 | #2501 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
708 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)(8,10,11,9);(4,5,6,7)(14,18)(16,19);(2,3)(8,9,11,10)(12,14)(13,16)(15,18)(17,19);(5,7)(9,10)(12,13,15,17)(14,16,18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10848565, #10848568, #10850469, … |
| maximal normal subgroup of | #10848565, #10848568, #10850469, … |
| maximal quotient of | #10717170, #10717416, #10718306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15435);