Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C17×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G | |
218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,24,26,38)(20,29)(21,34)(25,40,41,46)(27,39,28,35)(30,45,42,37)(32,48)(33,44)(36,47)(43,49);(18,19)(20,30)(21,35)(22,27)(23,28)(24,36)(25,33)(26,31)(29,41)(32,45)(34,38)(37,48)(39,47)(40,49)(42,44)(43,46);(18,20,31,44)(19,25,26,41)(21,32,47,48)(22,33,23,29)(24,37,38,45)(27,30,28,42)(34,49,36,43)(35,40,39,46)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 253);