Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#2166 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | #432 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#97 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,4,10,5,11,18,27,6,9,17,25,19,8,22,24)(3,12,14,26,15,23,29,32,13,7,16,21,28,20,30,31)(33,34,35);(1,3)(2,7)(4,16)(5,15)(6,13)(8,23)(9,12)(10,26)(11,20)(14,17)(18,30)(19,28)(21,25)(22,29)(24,31)(27,32);(2,8,9,11)(3,13)(4,17)(7,20,12,23)(10,27,25,24)(15,28)(18,22)(21,32,26,31)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5457);