Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C23 · Schur multiplier (H2) = C23 |
| direct product | C23×#1826 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C23 | C2×D4 |
| commutator | C2×C4 | C22×C4×C23 |
| Frattini, cosocle | C22×C4 | C23×C23 |
| socle | C22×C23 | C2×C22⋊C4 |
| prime radical | C24×C23 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C23 ◁ C23×C4×C23 ◁ G | |
306 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 736 |
A minimal faithful permutation representation — 3 generators on 39 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)(26,29)(27,28)(30,31)(32,33,35,38,36,34,39,37);(25,28)(29,31);(24,26,27,30)(25,29,28,31)(32,34)(33,36)(35,37)(38,39)874 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#8360, C23×#8362, C23×#8363, … |
| maximal normal subgroup of | C23×#8360, C23×#8362, C23×#8363, … |
| maximal quotient of | C23×#8706, C23×#8707, C23×#8708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 558);