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×#7167 |
| 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 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4)(5,6,8,11,9,12,7,10)(13,14,16,21,18,23,28,26,19,24,17,22,27,15,20,25)(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);(6,10,12,11)(13,15,17,26,18,14,20,22,19,23,16,25,27,24,28,21)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 155);