Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C8×#101×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C4×C22⋊C4 |
| prime radical | C23×C53 | C2×C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55,56,58,57,59,60,61);(62,63,65,69,66,68,74,76)(64,70,71,77,72,67,73,75);(62,64,66,72)(63,67,68,70)(65,73,74,71)(69,77,76,75)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1151);