Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C2×C42 |
| direct product | C53×#32678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C4×Q8×C53 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,6)(3,7,4,8)(9,10,13,15)(11,14,12,16)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31)(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);(2,6)(3,4)(9,11,13,12)(10,14,15,16)(17,19,21,27)(18,22,23,30)(20,25,28,26)(24,29,32,31);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(18,23)(22,30)(24,32)(29,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25666);