Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8068 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C22×C42×C53 ◁ G | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14,13,11,16,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(2,6,8,7)(9,11)(10,13)(12,15)(14,16);(1,3,5,4)(2,7,8,6)(9,12,13,16)(10,14,11,15)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1056);