Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13024 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #447 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1879 |
| prime radical | C53×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,13,7,14,16,12,8,15,25,32,24,31,5,11)(3,10,18,28,19,29,4,21,20,30,23,9,22,26,17,27)(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);(1,3,7,22,8,20,24,19)(2,9,14,21,15,28,31,27)(4,16,18,25,17,5,23,6)(10,12,26,32,30,11,29,13);(1,4)(2,10)(3,16)(5,20)(6,19)(7,18)(8,17)(9,12)(11,28)(13,27)(14,26)(15,30)(21,32)(22,25)(23,24)(29,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6012);