Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C53×#494 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
3262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(3,4);(6,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(5,6,7,8)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53618);