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) = C2×C4 |
| direct product | C53×#12670 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C23×C53 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,9,14,10,15,8,13,11,16,23,31,22,30,24,32)(7,17,18,12,20,27,25,28,21,33,35,29,26,36,19,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7)(6,12)(8,18)(9,25)(10,26)(11,21)(13,27)(14,33)(15,34)(16,29)(17,31)(19,23)(20,22)(24,35)(28,30)(32,36);(5,8,22,9,11,24,10,23)(6,13,30,14,16,32,15,31)(7,18,20,25,21,35,26,19)(12,27,28,33,29,36,34,17)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5658);