Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19228 |
| 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 | |
584 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)(3,10)(4,9)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18,22,20)(19,25,28,32)(21,29,30,24)(23,31,26,27)(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);(2,7)(8,15)(9,13)(10,16)(17,19,21,27,22,28,30,31)(18,23,24,25,20,26,29,32);(1,3)(2,8)(4,11)(5,12)(6,14)(7,15)(9,16)(10,13)(18,20)(23,26)(24,29)(25,32);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,16,15,10)(17,20,22,18)(19,26,28,23)(21,29,30,24)(25,27,32,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12216);