Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ #2424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#120 | C2 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2424 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7);(9,11)(12,13)(14,15);(2,3)(5,7)(8,9)(10,11)(12,13,14,15);(2,3)(4,5)(6,7)(12,13,14,15)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2424, C2×#82587, #10962717, … |
| maximal normal subgroup of | D3×#2424, C2×#82587, #10962717, … |
| maximal quotient of | #10717107, #10717372, #10717960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15163);