Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#12887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#106 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,22,12,26)(13,28)(14,31)(15,33)(16,21)(17,23,34,24)(18,30)(19,32)(20,29)(25,40,27,37)(35,39,36,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,4,7)(2,5,6,8)(9,11,13,23,15,25,30,38,16,12,14,24,29,27,32,39)(10,17,18,22,20,35,28,37,21,34,19,26,33,36,31,40);(2,6)(5,8)(9,12,13,24,15,27,30,39,16,11,14,23,29,25,32,38)(10,17,18,22,20,35,28,37,21,34,19,26,33,36,31,40)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5875);