Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C32 |
| direct product | H(3,𝔽3)×Q32 |
| unambiguous semidirect product | (C32×Q32)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×D8 |
| commutator, Frattini, cosocle | C3×C8 | C22×C32 |
| prime radical | C2×H(3,𝔽3) | D8 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C4×H(3,𝔽3) ◁ C8×H(3,𝔽3) ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27);(2,5,6)(4,8,9)(10,26,18,34)(11,27,19,35)(12,28,20,36)(13,29,21,37)(14,30,22,38)(15,31,23,39)(16,32,24,40)(17,33,25,41)121 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 609);