Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13201 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #390 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1955 |
| prime radical | C53×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C8×C53 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,12,6,13,25,22,8,14,26,31,5,11,16,28)(3,15,19,32,17,4,21,24,20,23,10,30,18,9,27,29)(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);(1,3,16,21,5,18,26,19,8,20,25,27,6,17,7,10)(2,9,28,32,11,23,31,29,14,4,22,30,13,15,12,24);(1,4,6,23,8,9,5,15)(2,10,11,21,14,19,13,27)(3,12,18,28,20,31,17,22)(7,24,25,30,26,29,16,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6189);