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 | #554×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 | |
228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 166 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:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14)(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);(2,4)(6,7)(9,11,13,16)(10,14,12,15);(1,3)(2,6)(4,7)(5,8);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,14,16,15)1775 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C71×#1622, C71×#1632, C71×#1643, … |
| maximal normal subgroup of | C71×#1622, C71×#1632, C71×#1643, … |
| maximal quotient of | C71×#2345, D4×C4⋊C4×C71, C71×#2349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4544, 235);