Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of #87648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Sylow | C72 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C7⋊D7 |
| commutator, socle, prime radical, cosocle | C2×C72 | C22 |
| Fitting | C4×C72 | C2 |
| hypercenter | C4 | C7⋊D7 |
| hypocenter | C72 | Q8 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4,8)(2,5,6,7)(16,17,19,21,22,20,18);(1,2,4,6)(3,7,8,5)(10,11)(12,13)(14,15)(17,18)(19,20)(21,22);(9,10,12,14,15,13,11)Polycyclic presentation:
2^3,7^2|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^6101 conjugacy classes — view the full character table.
| maximal subgroup of | #11716035, #11716038, #11716062, … |
| maximal normal subgroup of | #11716035, #11716038, #11716062, … |
| maximal quotient of | #11716068, C4⋊(C7⋊Dic7), C2×#87648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 28);