Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#106 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(2,4)(5,6)(7,15,17,31)(8,19)(9,12)(10,24)(11,16,26,32)(13,22)(14,23)(18,33,25,29)(20,27,35,30)(21,34)(28,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,10,25,9,17,23,18)(6,11,14,30,12,26,24,27)(8,20,22,16,21,35,36,32)(13,29,19,15,28,33,34,31);(1,2,3,4)(6,12)(11,26)(13,28)(14,24)(15,31)(19,34)(27,30)(29,33)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6172);