Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13098 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1898 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×2−1+4×C53 ◁ G | |
640 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)(3,12,9,5)(4,15,18,26)(6,10)(7,11)(8,17,24,22)(13,29)(14,28,21,20)(16,31)(19,30)(23,27)(25,32)(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);(1,3,5,2,7,9,12,11)(4,13,19,8,18,25,16,24)(6,21,20,27,23,14,28,10)(15,17,31,32,26,22,30,29);(1,4)(2,8)(3,13)(5,19)(6,22)(7,18)(9,25)(10,26)(11,24)(12,16)(14,31)(15,27)(17,23)(20,29)(21,30)(28,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6086);