Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#32691 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C4×Q8×C53 ◁ G | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,8)(3,7,4,6)(10,13)(11,15)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31)(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);(9,10,12,13)(11,16,15,14)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(18,23)(22,30)(24,32)(29,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25679);