Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #386 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,6,10,15)(4,7)(5,8)(9,12)(11,13,16,14)(17,18)(19,27,30,31)(20,24,23,29)(21,25)(22,26)(28,32)(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)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,21,24,22,30,32,29)(18,23,25,31,26,20,28,27);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,28)(21,27)(22,23)(25,30)(26,29)(31,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2403);