Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C23 · Schur multiplier (H2) = C22 |
| direct product | C23×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | #99 |
| commutator | C2×C4 | C42×C23 |
| Frattini, cosocle | C22×Q8 | C22×C23 |
| prime radical | C23×C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 736 |
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)(28,29,31,35,32,36,40,43)(30,34,37,42,38,33,39,41);(24,25,26,27)(29,33,36,34)(35,42,43,41)598 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#7404, C23×#7405, C23×#7409, … |
| maximal normal subgroup of | C23×#7404, C23×#7405, C23×#7409, … |
| maximal quotient of | C23×#7189, C23×#7206, C23×#7215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 126);