Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#12844 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C22 | C22×C16×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C23×C53 | C22×C8 |
| prime radical | C24×C53 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5832);