Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C53×#1765 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator | C4 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C4×D4 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C23×C8×C53 ◁ G | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 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);(7,8,11,15,12,16,10,13)(9,14,19,21,17,22,18,20);(3,4,5,6)(8,13,16,15)(9,17)(14,21,22,20)(18,19);(7,9,11,19,12,17,10,18)(8,14,15,21,16,22,13,20)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13407);