Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×C53×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C53 | D4 |
| commutator | C4 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C22×D4 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C53 ◁ C22×C42×C53 ◁ G | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(5,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13437);