Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#11485 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #99 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C2×M16 | C23×C53 |
| socle | C2×C53 | #1459 |
| prime radical | C53×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×C8×C53 ◁ C2×#131×C53 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,7,11,15,30,8,13,24,25,5,21,23,31)(3,14,17,29,16,9,20,26,18,28,19,32,22,27,4,10)(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)(2,9)(4,15)(5,22)(6,19)(7,16)(8,18)(10,25)(11,28)(12,26)(13,27)(14,21)(17,24)(20,23)(29,31)(30,32);(1,4,7,17,8,20,5,19)(2,10,11,29,13,26,21,32)(3,15,16,24,18,23,22,6)(9,25,28,31,27,12,14,30)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4473);