Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | D16 |
| commutator | C16 | C2×C4×C17 |
| Frattini, cosocle | C2×C16 | C22×C17 |
| socle | C2×C17 | #357 |
| prime radical | C17×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×C8×C17 ◁ C2×C16×C17 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,27,24,30,43,29)(20,31,33,60,36,61,35,67)(21,37,40,68,42,70,75,69)(23,44,39,28,46,56,74,58)(25,47,48,63,50,62,59,32)(26,51,53,76,55,78,54,41)(34,65,64,79,66,80,45,52)(38,57,71,77,73,81,72,49);(18,20)(19,25)(21,38)(22,33)(23,45)(24,36)(26,52)(27,48)(28,57)(29,59)(30,50)(31,37)(32,41)(34,39)(35,43)(40,71)(42,73)(44,49)(46,64)(47,51)(53,65)(54,80)(55,79)(56,77)(58,81)(60,68)(61,70)(62,78)(63,76)(66,74)(67,69)(72,75)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 149);