Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C34 · Schur multiplier (H2) = C2×C35 |
| direct product | C22×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C34 |
| commutator, Frattini, cosocle | C3 | C22×C34 |
| prime radical | C22×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(5,6,9,8,12,19,17,11,23)(7,13,24,16,22,18,15,10,20)(14,26,30,27,21,31,28,25,29);(3,4)(5,7,14)(6,10,21)(8,16,27)(9,18,29)(11,22,26)(12,13,25)(15,28,17)(19,20,30)(23,24,31);(7,15,16)(10,22,13)(14,27,28)(18,24,20)(21,25,26)(29,30,31);(6,11,12)(9,19,23)(10,22,13)(18,20,24)(21,26,25)(29,30,31)332 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 870);