Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13270 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #124 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1985 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10,5,11,14,28,7,12,23,29,22,9,26,25)(3,13,17,19,18,27,30,32,15,8,24,4,16,21,31,20)(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);(1,3,14,30,22,16,6,17,7,15,26,31,5,18,23,24)(2,8,25,32,9,27,29,19,12,13,28,20,11,21,10,4);(1,4,5,20,7,19,22,32)(2,3,9,16,12,15,11,18)(6,21,14,13,23,27,26,8)(10,17,25,31,29,24,28,30)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6258);