Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | SL(2,𝔽7) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | GL(3,𝔽2) |
| commutator, hypocenter | SL(2,𝔽7) | C2 |
| Frattini, socle, cosocle | C2 | C2×GL(3,𝔽2) |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C4 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,27,5,24,9,18,7)(2,4,21,12,16,15,10)(3,31,11,30,14,25,13)(6,8,26,19,23,20,17);(1,31,12,3,21,19,4,26,5,8,27,11)(2,28,16,6,29,23,7,32,24,13,22,30)(9,20,15,14)(10,25,18,17)22 conjugacy classes — view the full character table.
| maximal subgroup of | #13057023, #13057026, #13062002, … |
| maximal normal subgroup of | #13057023, #13057026, #13062002, … |
| maximal quotient of | C4×SL(2,𝔽7), C2×#10621465, #72209749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1047);