Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#21477 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #120 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | C4×#120 |
| prime radical | C53×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,12,7,11,24,10)(3,13,17,26,18,27,16,8)(4,19,22,23,21,9,20,5)(14,15,28,30,29,31,32,25)(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)(2,8)(4,14)(5,15)(6,17)(7,18)(9,25)(10,27)(11,26)(12,13)(16,24)(19,30)(20,32)(21,29)(22,28)(23,31);(1,4)(2,9)(3,14)(5,12)(6,22)(7,21)(8,25)(10,23)(11,19)(13,15)(16,32)(17,28)(18,29)(20,24)(26,30)(27,31);(1,5,6,19,7,23,24,9)(2,4,12,22,11,21,10,20)(3,15,17,30,18,31,16,25)(8,14,13,28,26,29,27,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 14465);