Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13264 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle, prime radical | C23×C53 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(13,14,18,22,16,15,19,24,17,21,23,26,25,20,27,28)(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);(2,4)(13,15,23,28,16,21,27,22,17,20,18,24,25,14,19,26);(5,7,8,12)(6,9,10,11)(13,16,17,25)(14,20,21,15)(18,19,23,27)(22,28,26,24)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6252);