Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#22265 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C53×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(2,4)(6,8)(10,11)(13,14,18,22,17,21,16,20)(15,19,23,26,24,27,25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,6,9,8)(7,11,12,10)(13,14,16,20,17,21,18,22)(15,19,25,28,24,27,23,26);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,15,17,24)(14,19,21,27)(16,25,18,23)(20,28,22,26);(5,8,9,6)(7,11,12,10)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15253);