Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C72 · Schur multiplier (H2) = C7 |
| direct product | C3×C72×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C72 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C72 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C72 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1176 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3)(4,5,6,7,8,9,10)(18,19,20,21)(22,25,24,23);(11,12,13,14,15,16,17)(18,22,20,24)(19,23,21,25)735 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 211);