Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11313 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #388 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1443 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
532 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)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,30,32,26)(14,24,21,25)(17,23,31,29)(19,28)(22,27)(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);(1,3,5,14,7,15,6,21)(2,8,20,25,10,12,11,24)(4,17,18,13,19,31,22,32)(9,26,28,23,27,30,16,29);(1,4)(2,9,10,27)(3,13)(5,18)(6,22)(7,19)(8,23,12,29)(11,16,20,28)(14,31)(15,32)(17,21)(24,26,25,30)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4301);