Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#12053 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle, prime radical | C22×C53 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C4×C16×C53 ◁ G | |
206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,8,5,9,7,15,6,10,13,16,14,3,11,12)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26)(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);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,3,4,12,5,2,7,8,6,9,13,15,14,10,11,16)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5041);