Finite Group DatabasebetaFinite Group Databasebeta| order | 26875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C43 · Schur multiplier (H2) = C5 |
| direct product | C43×#10597831 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C43 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C52×C43 |
| prime radical | C53×C43 | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C43 ◁ C52×C43 ◁ G | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 1075 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,4,8,15,3,5,10,16,22,7,11,19,17,23,14,13,9,18,24,21,6,12,20,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(2,5,11,13,6)(4,9,10,12,19)(8,16,17,18,20)2107 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26875, 8);