Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#62219 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C53×#529 ◁ G | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(18,23)(21,29)(24,31)(25,32)(26,30)(27,28)(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)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(19,20)(24,25)(27,28)(31,32);(17,18,22,26)(19,24,20,25)(21,30,29,23)(27,32,28,31);(3,10)(6,9)(11,16)(14,15)(17,19,22,20)(18,24,26,25)(21,27,29,28)(23,31,30,32);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,22,19)(18,25,26,24)(21,28,29,27)(23,32,30,31)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55207);