Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C17 |
| prime radical | C17×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C42×C17 ◁ G | |
228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,22)(20,24)(23,25)(26,27,31,35)(28,36,38,41)(29,34,30,33)(32,39,40,37);(20,24)(22,25)(26,28)(27,32)(29,39)(30,37)(31,38)(33,41)(34,36)(35,40);(18,19)(20,25)(21,23)(22,24)(26,28)(27,32)(29,37)(30,39)(31,38)(33,36)(34,41)(35,40)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 391);