Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#22898 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2358 |
| prime radical | C53×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,12,7,13,23,11)(3,14,17,25,5,10,16,8)(4,18,20,28,22,27,21,9)(15,30,32,26,19,24,31,29)(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);(1,3,6,17,7,5,23,16)(2,8,12,14,13,25,11,10)(4,15,20,32,22,19,21,31)(9,24,18,29,28,30,27,26);(1,4)(2,9)(3,15)(5,19)(6,20)(7,22)(8,24)(10,26)(11,27)(12,18)(13,28)(14,29)(16,31)(17,32)(21,23)(25,30);(1,5)(2,10)(3,7)(4,19)(6,16)(8,12)(9,26)(11,25)(13,14)(15,22)(17,23)(18,24)(20,31)(21,32)(27,30)(28,29)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15886);