Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C11 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D11 |
| commutator | C2×C11 | C22×C4 |
| Fitting | C11×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C22×C11 | C23 |
| hypercenter | C4⋊C4 | D11 |
| hypocenter | C11 | C4×Q8 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,8,10,11,9,7,5,3)(12,14,15,13)(18,22)(20,23);(16,18,19,22)(17,20,21,23);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,14)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^5,11|a^2=e,b^2=d,c^2=de,f^a=f^10,c^b=cd70 conjugacy classes — view the full character table.
| maximal subgroup of | #10622370, #10622371, #10622375, … |
| maximal normal subgroup of | #10622370, #10622371, #10622375, … |
| maximal quotient of | #10622259, #10622262, #10622265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 82);