Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C2×C53×#1369 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #106 |
| commutator | C8 | C22×C8×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C22×C53 | #334 |
| prime radical | C23×C53 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 87 points:
(1,2)(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);(3,4,7,13,9,14,24,30,10,15,23,12,8,26,6,22)(5,16,18,27,20,31,17,28,21,29,25,34,19,33,32,11);(3,5,6,17,8,19,23,18,10,21,24,32,9,20,7,25)(4,11,12,16,14,27,22,31,15,28,13,29,26,34,30,33)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4989);