Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#12982 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #120 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1869 |
| prime radical | C53×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,11,5,10,14,27,6,12,23,30,22,15,31,32)(3,13,17,4,9,8,26,21,18,28,24,20,16,25,29,19)(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);(1,3,14,29,22,16,7,24,6,18,31,26,5,9,23,17)(2,8,27,4,15,13,11,19,12,25,32,20,10,28,30,21);(1,4)(2,9)(3,15)(5,21)(6,20)(7,25)(8,23)(10,18)(11,29)(12,16)(13,14)(17,27)(19,22)(24,32)(26,30)(28,31)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5970);