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×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C23 | #121 |
| commutator | C2×C4 | C22×C4×C23 |
| Frattini, cosocle | C22×C4 | C23×C23 |
| socle | C2×C23 | #390 |
| prime radical | C2×C23×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C23 ◁ C22×C4×C23 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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) = 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)(26,30,35,44)(27,37)(28,40)(29,32)(31,38)(33,41)(34,46,47,55)(36,50,49,43)(39,51)(42,53,54,48)(45,52);(24,26)(25,30,32,44)(27,34)(28,36)(29,35)(31,42,45,54)(33,43,40,50)(37,46,51,55)(38,48)(39,47)(41,49)(52,53);(24,27,28,38,29,39,41,52)(25,31,33,37,32,45,40,51)(26,34,36,48,35,47,49,53)(30,42,43,46,44,54,50,55)736 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#13016, C23×#13021, C23×#13030, … |
| maximal normal subgroup of | C23×#13016, C23×#13021, C23×#13030, … |
| maximal quotient of | C23×#9667, C23×#9668, C23×#9669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 708);