Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13164 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #121 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1939 |
| prime radical | C23×C53 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,10,5,12,4,20,8,13,21,27,6,11,22,26)(3,14,17,29,18,25,15,30,19,28,31,32,16,9,23,24)(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);(1,3,7,17,5,18,4,15,8,19,21,31,6,16,22,23)(2,9,10,24,12,14,20,29,13,25,27,30,11,28,26,32);(1,4,6,7,8,22,5,21)(2,10,12,20,13,27,11,26)(3,15,16,17,19,23,18,31)(9,24,14,29,25,30,28,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6152);