Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#24422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,7,10,6,4,15,18)(3,8,14,11,13,16,22,26)(5,9,20,24,19,17,21,25)(12,23,29,32,28,31,30,27)(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);(1,3,6,13)(2,8,4,16)(5,12,19,28)(7,22,15,14)(9,23,17,31)(10,26,18,11)(20,30,21,29)(24,27,25,32);(1,4)(2,6)(3,11)(5,17)(7,10)(8,14)(9,19)(12,27)(13,26)(15,18)(16,22)(20,24)(21,25)(23,30)(28,32)(29,31);(1,5)(2,9)(3,12)(4,17)(6,19)(7,21)(8,23)(10,25)(11,27)(13,28)(14,30)(15,20)(16,31)(18,24)(22,29)(26,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17410);