Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#7174 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #106 |
| commutator | C8 | C4×C8×C53 |
| Frattini, cosocle | C2×C4×C8 | C22×C53 |
| socle | C22×C53 | #340 |
| prime radical | C23×C53 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,4,7,6,8,5,3)(9,10,13,17,15,20,23,19)(11,21,16,14,22,24,18,12)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,3,5,8,6,7,4,2)(9,11,14,10,13,18,24,19,15,22,12,20,23,16,21,17)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 162);