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 | C3×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
5796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,25)(13,26)(15,21)(16,22)(17,23)(18,24)(19,28)(20,30)(27,32)(29,31)(33,34,35);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,27)(16,28)(17,29)(18,30)(21,31)(22,25)(23,32)(24,26);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,17,18,15)(7,19,11,25)(9,23,24,21)(13,29,30,27)(20,32,26,31);(1,5,6,18)(2,9,10,24)(3,13,14,30)(4,15,16,17)(7,20,11,26)(8,21,22,23)(12,27,28,29)(19,31,25,32);(3,14)(7,11)(12,28)(13,30)(19,25)(20,26)(27,29)(31,32);(2,10)(7,11)(8,22)(9,24)(19,25)(20,26)(21,23)(31,32)195 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20161);