Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C23 · Schur multiplier (H2) = C22 |
| direct product | C23×#1387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | #102 |
| commutator | C8 | C42×C23 |
| Frattini, cosocle | C22×C8 | C22×C23 |
| prime radical | C23×C23 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C22×C8×C23 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 1472 |
A minimal faithful permutation representation — 2 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)(24,25,26,27)(29,33)(30,37)(31,38)(34,36)(35,42)(39,43)(40,41);(28,29,30,34,38,40,43,41,32,36,39,33,31,35,37,42)736 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C23×#1387, C23×#12569, C4×C23×#361, … |
| maximal normal subgroup of | C2×C23×#1387, C23×#12569, C4×C23×#361, … |
| maximal quotient of | C23×#7159, C23×#7176, C23×#7178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 119);