Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C23 · Schur multiplier (H2) = C2×C4 |
| direct product | C23×#1995 |
| 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 | C23×C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G | |
244 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 244 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) = 47 |
| transitive degree | μt(G) = 736 |
A minimal faithful permutation representation — 3 generators on 47 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)(26,30)(27,28)(29,31)(32,33,36,39,37,40,35,45)(34,41,44,46,42,47,43,38);(25,28)(29,30)(32,34)(33,38,40,46)(35,43)(36,44)(37,42)(39,41,45,47);(24,26,27,31)(25,29,28,30)(34,42)(38,46)(41,47)(43,44)736 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#7129, C23×#7360, C23×#8494, … |
| maximal normal subgroup of | C23×#7129, C23×#7360, C23×#8494, … |
| maximal quotient of | C23×#9872, C23×#9873, C23×#9874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 727);