Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12792 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #121 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1797 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,9,7,11,3,14,8,13,16,26,5,12,17,25)(4,18,21,32,20,28,31,24,22,29,23,30,19,10,15,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,6,8,17,7,16)(2,9,11,14,13,26,12,25)(4,15,19,23,22,31,20,21)(10,24,18,30,28,27,29,32);(1,4)(2,10)(3,15)(5,19)(6,23)(7,20)(8,22)(9,24)(11,18)(12,29)(13,28)(14,30)(16,21)(17,31)(25,32)(26,27)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5780);