Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of #87640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C72 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D72 |
| commutator, Fitting, socle, cosocle | C2×C72 | C22 |
| hypocenter | C72 | D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(6,7)(8,9)(10,11)(12,13,15,17,18,16,14);(2,4)(5,6,8,10,11,9,7)(13,14)(15,16)(17,18)Polycyclic presentation:
2^3,7^2|b^2=c,b^a=bc,e^a=e^6,d^b=d^647 conjugacy classes — view the full character table.
| maximal subgroup of | #11716100, #11716102, #11716103, … |
| maximal normal subgroup of | #11716100, #11716102, #11716103, … |
| maximal quotient of | #11716033, #11716035, #11716036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 21);