Finite Group DatabasebetaFinite Group Databasebeta| order | 2752 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C43 · Schur multiplier (H2) = C24 |
| direct product | C43×#394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C43 | C23 |
| commutator, Frattini, cosocle | C23 | C23×C43 |
| prime radical | C22×D4×C43 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C43 ◁ G | |
238 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 238 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) = 55 |
| transitive degree | μt(G) = 1376 |
A minimal faithful permutation representation — 3 generators on 55 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)(49,51)(52,53)(54,55);(45,47)(49,51)(53,54);(44,45)(46,47)(48,49)(50,51)(52,54,55,53)946 conjugacy classes — view the full character table.
| maximal subgroup of | C43×#1345, C43×#1597, C43×#1598, … |
| maximal normal subgroup of | C43×#1345, C43×#1597, C43×#1598, … |
| maximal quotient of | C43×#1438, C43×#1439, C43×#1440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2752, 75);