Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#7151 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | D4 |
| commutator | C4 | C4×C16×C53 |
| Frattini, cosocle | C2×C4×C8 | C22×C53 |
| socle, prime radical | C23×C53 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 2 generators on 81 points:
(6,10)(7,8)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2,3,4)(5,6,8,11,9,12,7,10)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 139);