Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #442 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6)(7,17,21,33)(8,23,22,35)(9,26)(10,28)(11,16)(12,20,30,27)(13,25,32,34)(14,18)(15,29)(19,31)(24,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,4)(5,7,18,25,29,8,9,27,11,21,19,34,10,22,24,20)(6,12,26,35,28,32,14,33,16,30,36,23,15,13,31,17);(5,8,24,34,10,7,19,27,11,22,9,25,29,21,18,20)(6,13,31,35,15,30,36,17,16,32,14,23,28,12,26,33)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6238);