Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
5992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,19);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16);(4,12)(7,14)(10,16)(13,15);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)195 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20160);