Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#12667 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #117 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1770 |
| prime radical | C23×C53 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×M32×C53 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,8,5,10,16,24,6,9,17,23,18,25,30,32)(3,11,14,27,15,12,26,28,13,7,19,21,29,22,20,31)(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)(2,7,9,11)(4,14)(5,15)(6,13)(8,21,23,27)(10,22,25,12)(16,26)(17,19)(18,29)(20,30)(24,31,32,28);(1,2,4,8,5,10,16,24,6,9,17,23,18,25,30,32)(3,12,19,31,15,7,20,27,13,22,14,28,29,11,26,21)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5655);