Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#24473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,10,7,11,5,21)(3,12,14,8,16,24,15,25)(4,17,19,28,20,27,22,9)(13,30,31,26,32,23,18,29)(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)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,21)(13,19)(17,23)(20,31)(22,32)(27,30)(28,29);(1,4)(2,9)(3,13)(5,19)(6,22)(7,20)(8,23)(10,27)(11,28)(12,29)(14,18)(15,31)(16,32)(17,21)(24,26)(25,30);(1,2,7,11)(3,8,16,25)(4,9,20,28)(5,21,6,10)(12,14,24,15)(13,23,32,30)(17,22,27,19)(18,26,31,29)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17461);