Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#24809 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,11,7,4,17,10)(3,12,14,8,16,24,15,18)(5,20,21,9,22,25,31,19)(13,27,28,26,30,23,29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,6,14,7,16,17,15)(2,8,10,12,4,18,11,24)(5,13,21,28,22,30,31,29)(9,23,20,26,19,27,25,32);(1,4)(2,7)(3,12)(5,19)(6,11)(8,15)(9,22)(10,17)(13,26)(14,18)(16,24)(20,31)(21,25)(23,29)(27,28)(30,32);(1,5,17,31,7,22,6,21)(2,9,11,25,4,19,10,20)(3,13,15,29,16,30,14,28)(8,23,24,32,18,27,12,26)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17797);