Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C27 |
| direct product | C2×C53×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | #386 |
| prime radical | C2×C53×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 71 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);(56,57,59,63,60,62,68,70)(58,64,66,71,65,61,67,69);(56,57,59,63,60,62,68,70)(58,61,67,71,65,64,66,69);(56,58)(57,61)(59,67)(60,65)(62,64)(63,71)(66,68)(69,70)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13951);