Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13099 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1898 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×H(5,𝔽2)×C53 ◁ G | |
666 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)(3,13,16,29)(4,10,5,18)(6,11)(7,12)(8,14)(9,17,27,20)(15,23,21,32)(19,25)(22,26)(24,30)(28,31)(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);(1,3,5,19,7,16,4,14)(2,8,10,13,12,25,18,29)(6,21,20,30,22,15,17,31)(9,23,11,28,27,32,26,24);(1,4,7,5)(2,9,12,27)(3,14,16,19)(6,20,22,17)(8,23,25,32)(10,11,18,26)(13,28,29,24)(15,31,21,30)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6087);