Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of #5997 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C7 |
No aliases for this group.
| derived, Sylow | C72 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #5997 |
| commutator, hypocenter | C72 | C8 |
| Fitting, socle, prime radical | C2×C72 | C4 |
| cosocle | C7⋊Dic7 | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,5,6,10)(4,8,9,13)(7,11,12,14)(15,16,17,19,18,20,21,22);(1,3,7,13,8,12,6)(2,9,5,14,11,10,4)Polycyclic presentation:
2^3,7^2|a^2=b,b^2=c,d^a=d^4e,e^a=d^4e^3,d^b=d^6,e^b=e^632 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 17);