Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of #87658 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, Sylow | C72 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, prime radical, cosocle | C7⋊D7 | C22 |
| Fitting, socle, hypocenter | C72 | Q8 |
| LCS | G ▷ C7⋊D7 ▷ C72 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,3,4,10)(2,27,26,13)(5,17,25,28)(6,22,18,14)(7,15)(8,11,21,12)(9,16)(19,20,23,24);(1,2)(3,5)(4,9,11,18)(6,15,17,19)(7,16,20,13)(8,22,24,27)(10,25,21,14)(12,26,23,28)Polycyclic presentation:
2^3,7^2|a^2=c,b^2=c,b^a=bc,d^a=d^4e,e^a=d^4e^3,d^b=d^3e^5,e^b=d^5e^4,d^c=d^6,e^c=e^6| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 8 | 8 | 8 | 8 | 8 | 8 | 49 | 98 | 98 | 98 |
| order | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 2 | 4 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 |
| χ6 | 8 | 1 | 1 | 1 | ζ+3ζ3+3ζ4+ζ6 | 3ζ+ζ2+ζ5+3ζ6 | 3ζ2+ζ3+ζ4+3ζ5 | 0 | 0 | 0 | 0 |
| χ7 | 8 | 1 | 1 | 1 | 3ζ+ζ2+ζ5+3ζ6 | 3ζ2+ζ3+ζ4+3ζ5 | ζ+3ζ3+3ζ4+ζ6 | 0 | 0 | 0 | 0 |
| χ8 | 8 | 1 | 1 | 1 | 3ζ2+ζ3+ζ4+3ζ5 | ζ+3ζ3+3ζ4+ζ6 | 3ζ+ζ2+ζ5+3ζ6 | 0 | 0 | 0 | 0 |
| χ9 | 8 | ζ+3ζ3+3ζ4+ζ6 | 3ζ+ζ2+ζ5+3ζ6 | 3ζ2+ζ3+ζ4+3ζ5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| χ10 | 8 | 3ζ+ζ2+ζ5+3ζ6 | 3ζ2+ζ3+ζ4+3ζ5 | ζ+3ζ3+3ζ4+ζ6 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| χ11 | 8 | 3ζ2+ζ3+ζ4+3ζ5 | ζ+3ζ3+3ζ4+ζ6 | 3ζ+ζ2+ζ5+3ζ6 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 7 (ζ7 = 1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 38);