Finite Group DatabasebetaFinite Group Databasebeta| order | 4544 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C71 · Schur multiplier (H2) = C24 |
| direct product | #556×C71 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C71 |
| prime radical | C2×D4×C71 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ G | |
212 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 2272 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2)(3,5)(4,7)(6,8)(11,15)(13,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,3,6,4)(2,7,8,5)(9,10,12,14)(11,13,15,16);(1,4,6,3)(2,7,8,5)(9,11)(10,13)(12,15)(14,16);(1,5,6,7)(2,3,8,4)1775 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C71×#1628, C71×#1638, C71×#1650, … |
| maximal normal subgroup of | C71×#1628, C71×#1638, C71×#1650, … |
| maximal quotient of | C71×#2347, C71×#2354, C71×#2355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4544, 237);