Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#24297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,8,12)(3,7,11,6)(4,13,19,25)(5,20,23,14)(9,17,15,18)(10,22,26,21)(16,24,31,28)(27,30,32,29)(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);(1,3,6,2,8,11,7,12)(4,15,17,13,19,9,18,25)(5,14,21,10,23,20,22,26)(16,28,29,27,31,24,30,32);(1,4)(2,9)(3,13)(5,16)(6,17)(7,18)(8,19)(10,24)(11,25)(12,15)(14,27)(20,32)(21,29)(22,30)(23,31)(26,28);(1,5,7,22,8,23,6,21)(2,10,3,14,12,26,11,20)(4,16,18,30,19,31,17,29)(9,24,13,27,15,28,25,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17285);