Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#23043 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2361 |
| prime radical | C53×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,7,13)(3,14)(4,21,10,5)(6,11,15,12)(8,19)(9,22,27,23)(16,17)(18,26)(20,29)(24,25)(28,31)(30,32)(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);(1,3,15,18,7,20,6,19)(2,8,12,14,13,26,11,29)(4,16,23,25,10,28,22,32)(5,24,27,31,21,30,9,17);(1,4)(2,9)(3,16)(5,12)(6,22)(7,10)(8,17)(11,21)(13,27)(14,24)(15,23)(18,25)(19,32)(20,28)(26,31)(29,30);(1,5,6,9,7,21,15,27)(2,10,11,23,13,4,12,22)(3,17,18,24,20,31,19,30)(8,25,14,28,26,32,29,16)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16031);