Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | SO(3,𝔽7) |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽7) | C2 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C2 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,4,14)(2,9,16)(3,7,15)(8,12,10)(17,19,27)(18,22,20)(21,28,32)(23,29,26);(1,32,2,29,13,20,14,24,12,19,7,22,6,26,8,30)(3,21,5,17,4,23,15,25,16,27,11,28,10,18,9,31)16 conjugacy classes — view the full character table.
| maximal subgroup of | #13062003, #13062004, C2×#10621463, … |
| maximal normal subgroup of | #13062003, #13062004, C2×#10621463, … |
| maximal quotient of | #13057024, C2×#10621463, C3×#10621463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1045);