Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C37 · Schur multiplier (H2) = C35 |
| direct product | C37×#10624564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C37 | C33 |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| prime radical | C32×H(3,𝔽3)×C37 | C3 |
| LCS | G ▷ C33 ▷ C1 | |
| UCS | C1 ◁ C33×C37 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 82 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 3 generators on 82 points:
(1,2,4)(3,6,8)(5,9,7)(19,20,23)(21,25,31)(22,26,30)(24,32,29)(27,34,39)(28,35,38)(33,40,37)(36,42,44)(41,45,43)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(10,11,13)(12,15,17)(14,18,16)(20,24,26)(23,30,32)(25,33,35)(31,38,40)(34,41,42)(39,44,45);(2,5,6)(4,8,9)(11,14,15)(13,17,18)(19,21,27,22,28,36,29,37,43)(20,25,34,26,35,42,24,33,41)(23,31,39,30,38,44,32,40,45)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 124);