Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #447 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C24×C53 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×M16×C53 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6,9,15,23,31,24,33,11,13,8,14,10,16,26,35)(7,17,20,30,27,34,22,12,19,29,18,36,21,32,25,28)(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);(1,3,4,2)(5,7)(6,12)(8,22)(9,25)(10,27)(11,19)(13,28)(14,32)(15,34)(16,36)(17,33)(18,24)(20,26)(21,23)(29,35)(30,31);(6,13)(8,9)(12,28)(16,31)(18,20)(22,25)(24,26)(30,36)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5988);