Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13283 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,22)(17,27)(18,28)(19,24)(20,23)(21,26)(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);(1,3,4,2)(13,15,16,24,18,21,26,28,19,22,25,14,20,27,17,23);(5,7,8,12)(6,9,10,11)(13,16,18,26,19,25,20,17)(14,21,23,22,24,27,28,15)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6271);