Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#10249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18)(19,23)(20,22)(21,25)(24,28)(26,31)(27,32)(29,30)(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)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(18,22)(20,28)(23,31)(24,25)(26,29)(30,32);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,19,21,27)(18,23,25,32)(20,29,28,26)(22,30,24,31)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3237);