Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#61067 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C4×D4×C53 ◁ G | |
2778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 73 points:
(1,2)(3,4)(5,6,11,14)(7,15,17,19)(8,13,10,9)(12,18,20,16)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,7)(6,12)(8,16)(9,15)(10,18)(11,17)(13,19)(14,20);(2,4)(5,8,11,10)(6,13,14,9)(7,16,17,18)(12,19,20,15);(5,9)(6,8)(7,12)(10,14)(11,13)(15,18)(16,19)(17,20);(7,17)(12,20)(15,19)(16,18)2915 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54055);