Finite Group DatabasebetaFinite Group Databasebeta| order | 2752 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C43 · Schur multiplier (H2) = C23 |
| direct product | C43×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C43 | C2×D4 |
| commutator | C4 | C22×C4×C43 |
| Frattini, cosocle | C2×C4 | C23×C43 |
| socle | C2×C43 | C2×C22⋊C4 |
| prime radical | C2×#41×C43 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C43 ◁ C22×C4×C43 ◁ G | |
154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 688 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)(44,45,48,51,49,52,47,56)(46,53,55,50,54,59,57,58);(44,46)(45,50)(47,57)(48,55)(49,54)(51,59)(52,58)(53,56);(46,54)(50,58)(53,59)(55,57)946 conjugacy classes — view the full character table.
| maximal subgroup of | C43×#1794, C43×#1795, C43×#1796, … |
| maximal normal subgroup of | C43×#1794, C43×#1795, C43×#1796, … |
| maximal quotient of | C43×#1477, C43×#1478, C43×#1479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2752, 102);