Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #124 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #397 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
546 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)(7,15,19,29)(8,20)(9,13)(10,14)(11,18,27,24)(12,23)(16,25)(17,30,21,35)(22,34)(26,32,33,36)(28,31)(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);(5,7,16,18,10,19,9,24)(6,11,25,15,14,27,13,29)(8,21,31,36,23,17,22,32)(12,26,34,30,20,33,28,35);(5,8,9,22,10,23,16,31)(6,12,13,28,14,20,25,34)(7,17,18,32,19,21,24,36)(11,26,15,30,27,33,29,35)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4308);