Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C16×C53 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2)(3,4)(5,6,11,14,10,15,26,27,12,17,25,31,9,16,8,24)(7,18,22,32,21,30,34,36,23,29,33,35,20,13,19,28)(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);(5,7,8,19,9,20,25,33,12,23,26,34,10,21,11,22)(6,13,27,32,16,29,14,28,17,30,24,35,15,18,31,36);(2,4)(5,8,9,25,12,26,10,11)(6,14,15,27,17,31,16,24)(7,19,20,33,23,34,21,22)(13,28,18,32,30,36,29,35)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6112);