Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#23042 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D42 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2361 |
| prime radical | C53×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,8,13)(3,14)(4,5,11,23)(6,12,7,19)(9,24)(10,25,22,20)(15,30)(16,29)(17,28)(18,27)(21,31)(26,32)(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);(1,3,7,17,8,18,6,24)(2,9,19,14,13,28,12,27)(4,15,25,26,11,29,20,31)(5,21,22,30,23,32,10,16);(1,4)(2,10)(3,15)(5,19)(6,20)(7,25)(8,11)(9,16)(12,23)(13,22)(14,21)(17,26)(18,29)(24,31)(27,32)(28,30);(1,5,6,10,8,23,7,22)(2,11,12,25,13,4,19,20)(3,16,17,21,18,30,24,32)(9,26,14,29,28,31,27,15)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16030);