Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#61105 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C2×C53 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
2620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 5 generators on 85 points:
(2,9)(3,13)(6,7)(11,25)(12,22)(14,28)(15,24)(16,27)(18,23)(20,21)(29,32)(30,31)(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);(1,2)(3,5)(4,16)(6,22)(7,9)(8,12)(10,15)(11,18)(13,21)(14,29)(17,30)(19,25)(20,24)(23,27)(26,28)(31,32);(1,3)(2,10)(4,14)(5,12)(6,13)(7,24)(8,15)(9,21)(11,26)(16,32)(17,27)(18,30)(19,31)(20,22)(23,28)(25,29);(1,4)(2,11)(3,14)(5,17)(6,23)(7,18)(8,19)(9,25)(10,26)(12,27)(13,28)(15,31)(16,22)(20,32)(21,29)(24,30);(1,5,8,10)(2,3,12,15)(4,17,19,26)(6,20,7,21)(9,13,22,24)(11,14,27,31)(16,30,25,28)(18,29,23,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54093);