Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C29 |
| direct product | C23×#108×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | D4 |
| commutator | C4 | C24×C4×C53 |
| Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C24×C53 | C4⋊C4 |
| prime radical | C25×C53 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ C24×C4×C53 ◁ G | |
2846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 75 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(3,4);(5,6);(7,8,11,14,12,15,10,20)(9,16,18,21,19,22,17,13);(7,9,10,17,12,19,11,18)(8,13,14,16,15,21,20,22)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53098);