Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#22039 |
| 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×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,21,18,15,19,23)(16,20,25,27,24,22,26,28)(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);(5,7)(6,9)(8,12)(10,11)(13,15,17,23,18,14,19,21)(16,22,25,28,24,20,26,27);(2,4)(7,12)(9,11)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(6,10)(9,11)(14,15)(16,24)(21,23)(25,26)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15027);