Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#7182 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #102 |
| commutator | C8 | C4×C8×C53 |
| Frattini, cosocle | C2×C4×C8 | C22×C53 |
| socle | C22×C53 | D4⋊C8 |
| prime radical | C23×C53 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4)(5,6,8,13,10,14,7,16,11,15,18,20,9,19,17,12)(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);(6,12,14,13,15,16,19,20)(7,17)(8,18)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 170);