Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of #67369 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, derived | C32×C7 ⋊ C2×C3 |
| Hall | C7⋊H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ (C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C7 | C2×C3 |
| Frattini | C3 | C3⋊AGL(1,𝔽7) |
| socle | C3×C7 | C3×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5,6,3,4,7)(9,14,16,11,12,15)(10,13);(1,2,4,7,6,5,3)(8,9,11)(10,12,15)(13,16,14)Polycyclic presentation:
2,3^3,7|c^a=c^2,d^a=d^2,e^a=e^6,c^b=cd,e^b=e^219 conjugacy classes — view the full character table.
| maximal subgroup of | D7⋊((C3⋊D3)⋊C3), C2×#67369, #11765620, … |
| maximal normal subgroup of | D7⋊((C3⋊D3)⋊C3), C2×#67369, ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3), … |
| maximal quotient of | #10625400, C2×#67369, #11765620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 22);