Finite Group DatabasebetaFinite Group Databasebeta| order | 2752 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C43 · Schur multiplier (H2) = C25 |
| direct product | C43×#548 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C43 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C43 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C43 ◁ G | |
290 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 290 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) = 55 |
| transitive degree | μt(G) = 1376 |
A minimal faithful permutation representation — 4 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)(45,47)(49,52)(53,54);(48,49,51,52)(50,54,55,53);(44,45)(46,47);(48,50,51,55)(49,53,52,54)1075 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C43×#3265, C43×#3269, C43×#3270, … |
| maximal normal subgroup of | C43×#3265, C43×#3269, C43×#3270, … |
| maximal quotient of | Q8×C22⋊C4×C43, C43×#2341, C43×#2351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2752, 229);