Finite Group DatabasebetaFinite Group Databasebeta| order | 294 |
| isoclinism | isoclinic class of #64423 |
| automorphism | Aut(G) = AGL(1,𝔽7)2 (order 1764) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C72 ⋊ C2×C3 |
| Hall | #3861 ⋊ C2 |
| Hall | C7×D7 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 14 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,11,7,5,6,12)(4,13,10,8,9,14);(2,12,11,8,5,13,14)Polycyclic presentation:
2,3,7^2|d^a=d^6,c^b=c^4,d^b=d^417 conjugacy classes — view the full character table.
| maximal subgroup of | #64423⋊C2, C2×#64423, #64418⋊C3, … |
| maximal normal subgroup of | #64423⋊C2, C2×#64423, ΓL(1,𝔽8)×AGL(1,𝔽7), … |
| maximal quotient of | #10596873, C2×#64423, #11725076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(294, 12);