Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#10794 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | C22⋊#95 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×M16×C53 ◁ G | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,12,7,10,6,11)(3,8,15,23,14,24,20,22)(4,16,19,25,17,26,18,9)(13,27,29,32,30,31,28,21)(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);(1,3,6,15,7,14,5,20)(2,8,12,22,10,24,11,23)(4,13,18,29,17,30,19,28)(9,21,16,31,25,32,26,27);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,21)(10,25)(11,26)(12,16)(14,30)(15,29)(20,28)(22,31)(23,27)(24,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3782);