Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#10950 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #386 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1437 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
904 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)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,27,28,32)(14,23,21,24)(17,30)(19,25)(22,29,26,31)(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);(1,3,6,14,7,15,5,21)(2,8,11,24,10,12,20,23)(4,13,19,29,17,28,18,31)(9,22,16,27,25,26,30,32);(1,4,5,18,7,17,6,19)(2,9,11,16,10,25,20,30)(3,13,21,31,15,28,14,29)(8,22,24,27,12,26,23,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3938);