Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C53 · Schur multiplier (H2) = C214 |
| direct product | C53×#63076 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C26 |
| commutator | C2 | C25×C4×C53 |
| Frattini, cosocle | C4 | C26×C53 |
| socle | C2×C53 | C25×C4 |
| prime radical | C2×C53×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ G | |
10796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 6 generators on 85 points:
(1,2,5,10,6,9,16,22)(3,7,13,19,12,18,25,29)(4,8,15,21,14,20,26,30)(11,17,24,28,23,27,31,32)(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)(2,7)(4,11)(5,13)(6,12)(8,17)(9,18)(10,19)(14,23)(15,24)(16,25)(20,27)(21,28)(22,29)(26,31)(30,32);(1,4)(2,8)(3,11)(5,15)(6,14)(7,17)(9,20)(10,21)(12,23)(13,24)(16,26)(18,27)(19,28)(22,30)(25,31)(29,32);(4,14)(8,20)(11,23)(15,26)(17,27)(21,30)(24,31)(28,32);(3,12)(7,18)(11,23)(13,25)(17,27)(19,29)(24,31)(28,32);(2,9)(7,18)(8,20)(10,22)(17,27)(19,29)(21,30)(28,32)6890 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 56064);