Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8731 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #117 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #381 |
| prime radical | C24×C53 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#376 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,4,7,5,8,3,6)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(21,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23);(1,3,5,4)(2,6,8,7)(9,11)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1719);