Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 384 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C3×C7×C16 |
| multiplicative | (ℤ/731)ˣ, (ℤ/791)ˣ, (ℤ/833)ˣ, (ℤ/1011)ˣ, (ℤ/1017)ˣ, (ℤ/1348)ˣ, (ℤ/1462)ˣ, (ℤ/1582)ˣ, (ℤ/1666)ˣ, (ℤ/2022)ˣ, (ℤ/2034)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8 | C22×C3×C7 |
| socle, prime radical | C22×C3×C7 | C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(19,20,21)(22,23,24,25,26,27,28);(17,18)672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 273);