Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#11988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#102 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C23×C53 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 77 points:
(2,6)(3,4)(7,8)(9,10,14,19,16,17,24,22,15,20,13,18,12,11,21,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2)(3,6)(4,8)(5,7)(9,11,15,17)(10,16,20,12)(13,19,14,18)(21,23,24,22);(1,3,5,4)(2,6,7,8)(9,12,15,16)(10,17,20,11)(13,24,14,21)(18,23,19,22)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4976);