Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of #67361 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived | C7×C9 ⋊ C2×C3 |
| Hall | #4427 ⋊ C2 |
| Sylow | C7 ⋊ AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9 | C2×C3 |
| Frattini | C3 | D3×ΓL(1,𝔽8) |
| socle | C3×C7 | C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,4)(5,7,6)(9,15,16,11,12,13)(10,14);(1,2,3,5,4,6,7)(8,12,11,10,16,13,14,9,15)Polycyclic presentation:
2,3^2,7,3|c^3=e,c^a=c^2e^2,e^a=e^2,c^b=ce,d^b=d^222 conjugacy classes — view the full character table.
| maximal subgroup of | #67356⋊C2, C2×#67361, C3×#67361, … |
| maximal normal subgroup of | #67356⋊C2, C2×#67361, C3×#67361, … |
| maximal quotient of | #10625391, C2×#67361, #11765604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 14);