Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13103 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1898 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×2−1+4×C53 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,12,8,13,6,11)(3,14,17,15,18,5,10,25)(4,19,22,9,23,27,21,28)(16,31,29,24,32,30,26,20)(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);(1,3,15,13)(2,8,18,25)(4,16,30,28)(5,11,6,10)(7,17,14,12)(9,23,32,31)(19,21,26,20)(22,29,24,27);(1,4,8,23)(2,9,13,28)(3,16,18,32)(5,24,14,20)(6,22,7,21)(10,29,17,26)(11,27,12,19)(15,30,25,31)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6091);