Finite Group DatabasebetaFinite Group Databasebeta| order | 4544 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C71 · Schur multiplier (H2) = C25 |
| direct product | C2×#137×C71 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C71 |
| prime radical | C2×#41×C71 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ G | |
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 2272 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(3,4)(5,11)(6,14)(7,9)(8,12)(10,16)(13,18)(15,17);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11);(4,9)(8,17)(10,14)(11,18)1562 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C71×#1882, C71×#1902, C71×#1905, … |
| maximal normal subgroup of | C71×#1882, C71×#1902, C71×#1905, … |
| maximal quotient of | C2×#419×C71, C2×#417×C71, C2×#428×C71, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4544, 255);