Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#1380 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 79 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(3,4,5,6)(12,17)(13,21)(14,23)(15,22)(18,19)(20,26)(24,25);(7,8,9,10)(11,12,14,18,15,19,13,17,16,20,21,25,22,24,23,26)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5543);