Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C37 · Schur multiplier (H2) = C9 |
| direct product | C37×#10624515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C37 | #6867 |
| commutator | C3×C9 | C3×C9×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| prime radical | C33×C37 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C33×C37 ◁ C32×C9×C37 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 118 points:
(1,2,6)(3,11,33,14,36,58,15,29,66)(4,16,22)(5,10,23)(7,24,42,26,59,49,27,55,47)(8,28,19)(9,32,20)(12,37,73,39,72,79,44,74,57)(13,40,53,43,68,54,17,35,21)(18,48,56)(25,60,38,62,81,41,63,70,46)(30,69,78)(31,65,51)(34,71,77,67,80,75,61,52,76)(45,64,50)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(1,3,12)(2,7,25,9,26,63,10,27,62)(4,17,46)(5,15,39)(6,21,52,23,54,71,32,53,80)(8,29,67,16,11,34,31,36,61)(13,41,51)(14,44,20)(18,49,76)(19,43,38)(22,55,73,28,59,79,65,24,57)(30,40,37,45,68,74,48,35,72)(33,70,56,66,60,64,58,81,69)(42,77,50)(47,75,78)2109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 75);