Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#17968 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C53×#3171 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#431 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(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);(4,12)(8,10)(13,14)(15,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(19,26)(23,30)(27,32)(29,31)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10956);