Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#10770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #381 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,9,14,10,13,23,28)(7,11,17,26,16,25,21,29)(8,18,19,30,20,12,22,27)(15,31,33,36,32,24,34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7)(6,11,13,25)(8,15)(9,21)(10,16)(12,24,18,31)(14,29,28,26)(17,23)(19,34)(20,32)(22,33)(27,36,30,35);(5,8,10,20)(6,12,13,18)(7,15,16,32)(9,22,23,19)(11,24,25,31)(14,30,28,27)(17,34,21,33)(26,36,29,35)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3758);