Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#12671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C23×C53 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,9,15,10,16,8,14,11,17,24,33,23,32,25,13)(7,18,19,12,21,29,26,30,22,34,36,31,27,28,20,35)(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);(1,2,3,4)(5,7)(6,12)(8,19)(9,26)(10,27)(11,22)(13,28)(14,29)(15,34)(16,35)(17,31)(18,33)(20,24)(21,23)(25,36)(30,32);(5,8,23,9,11,25,10,24)(6,13,32,33,17,14,16,15)(7,19,21,26,22,36,27,20)(12,28,30,18,31,29,35,34)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5659);