Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #459 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1802 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,11,8,13,5,23)(3,14,16,25,17,12,6,9)(4,18,21,30,22,32,19,29)(10,24,26,15,28,31,27,20)(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);(1,3,5,6,8,17,7,16)(2,9,11,14,13,25,23,12)(4,15,19,24,22,20,21,31)(10,18,26,30,28,32,27,29);(1,4)(2,10)(3,15)(5,19)(6,24)(7,21)(8,22)(9,18)(11,26)(12,29)(13,28)(14,30)(16,31)(17,20)(23,27)(25,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5810);