Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | #121×C53 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(17,18,22,24)(19,27,30,31)(20,25,23,32)(21,29,28,26)(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);(3,10)(7,14)(11,16)(13,15)(17,19,28,32,22,30,21,25)(18,23,29,31,24,20,26,27);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(18,24)(20,23)(26,29)(27,31);(1,3,5,10)(2,7,9,14)(4,11,12,16)(6,13,8,15)(17,20)(18,25)(19,29)(21,31)(22,23)(24,32)(26,30)(27,28)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12575);