Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C34 |
| direct product | C22×#6911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C22×C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C33 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(5,6,9)(7,13,18)(8,12,19)(10,20,16)(11,23,17)(14,25,28)(15,22,29)(21,31,27)(24,30,26);(3,4)(5,7,14,17,15,27,8,16,26)(6,10,21,11,13,24,12,22,25)(9,18,28,23,29,31,19,20,30);(7,15,16)(10,13,22)(14,26,27)(18,29,20)(21,25,24)(28,30,31)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 718);