Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#8189 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C2×D4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C4×C22⋊C4 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,25,22,26,20,23)(19,24,28,31,29,32,27,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,10,5,9,11,15)(2,6,8,12,7,13,14,16)(18,23,26,25)(24,30,32,31);(2,7)(3,9)(8,14)(10,15)(17,19,21,28,22,29,20,27)(18,24,25,31,26,32,23,30)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1177);