Finite Group DatabasebetaFinite Group Databasebeta| order | 38125 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C61 · Schur multiplier (H2) = C54 |
| direct product | C5×C61×H(3,𝔽5) |
| unambiguous semidirect product | C52⋊(C52×C61) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C61 | C52 |
| commutator, Frattini, cosocle | C5 | C53×C61 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C61 ◁ G | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 7625 |
A minimal faithful permutation representation — 3 generators on 91 points:
(1,2,3,4,5)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(6,7,9,13,20)(8,11,15,22,27)(10,16,25,30,26)(12,18,24,21,28)(14,23,29,19,17);(7,10,17,18,11)(9,14,15,16,24)(13,21,25,22,23)(20,27,28,29,30)8845 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38125, 12);