Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C23 · Schur multiplier (H2) = C25 |
| direct product | C23×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C23 | #120 |
| commutator | C23 | C22×C4×C23 |
| Frattini, cosocle | C22×C4 | C23×C23 |
| socle | C2×C23 | #379 |
| prime radical | C23×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C23 ◁ C22×C4×C23 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 736 |
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)(24,25,28,33,29,32,41,46)(26,34,37,47,36,30,40,43)(27,31,39,44,38,45,51,55)(35,42,50,53,49,48,52,54);(24,26)(25,30)(27,35)(28,40)(29,36)(31,42)(32,34)(33,47)(37,41)(38,49)(39,52)(43,46)(44,54)(45,48)(50,51)(53,55);(24,27)(25,31)(26,35)(28,39)(29,38)(30,42)(32,45)(33,44)(34,48)(36,49)(37,50)(40,52)(41,51)(43,53)(46,55)(47,54)736 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#13045, C23×#13046, C23×#13051, … |
| maximal normal subgroup of | C23×#13045, C23×#13046, C23×#13051, … |
| maximal quotient of | C23×#8316, C23×#8317, C23×#8348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 523);