Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C23 · Schur multiplier (H2) = C22×C4 |
| direct product | C23×#1974 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C23 | C2×D4 |
| commutator | C2×C4 | C22×C4×C23 |
| Frattini, cosocle | C22×C4 | C23×C23 |
| socle | C22×C23 | C4×D4 |
| prime radical | C2×C23×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C23 ◁ C22×C8×C23 ◁ G | |
300 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 736 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(25,27)(28,29,32,36,33,37,31,34)(30,38,40,43,41,42,39,35);(24,25,26,27)(29,34,37,36)(35,42,43,38);(28,30,31,39,33,41,32,40)(29,35,36,38,37,43,34,42)1012 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#7114, C23×#13011, C23×#13027, … |
| maximal normal subgroup of | C23×#7114, C23×#13011, C23×#13027, … |
| maximal quotient of | C23×#9655, C23×#9656, C23×#9657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 706);