Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#17353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C22 | C23×C8×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×#41×C53 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C4×Q8×C53 ◁ G | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,5,4,6,7,8)(9,10,14,19,15,18,12,17)(11,20,21,24,13,16,22,23)(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);(9,11,15,13)(10,16,18,20)(12,21,14,22)(17,23,19,24);(9,12,15,14)(10,17,18,19)(11,21,13,22)(16,23,20,24);(9,13,15,11)(10,16,18,20)(12,21,14,22)(17,24,19,23)4346 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10341);