Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#10804 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #117 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #382 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#117×C53 ◁ G | |
642 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)(7,15,18,28)(8,19,21,12)(9,13)(10,14)(11,17,24,32)(16,29)(20,34,35,25)(22,26)(23,30)(27,31)(33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,23,19,33)(13,15,26,28)(20,30)(21,31)(22,32)(25,27,34,29)(35,36);(5,8,9,20,10,21,22,35)(6,12,13,25,14,19,26,34)(7,16,17,30,18,31,32,36)(11,23,15,27,24,33,28,29)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3792);