Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×#41 ⋊ C2 |
| Sylow | C11 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C22⋊D11 |
| commutator, cosocle | C4×C11 | C23 |
| Frattini | C4 | C22×D11 |
| Fitting | C11×#41 | C2 |
| socle | C2×C11 | C2×D4 |
| hypercenter | #41 | D11 |
| hypocenter | C11 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 88 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,8,10,11,9,7,5,3)(12,13)(14,19)(15,17)(16,18);(13,17)(16,19);(2,3)(4,5)(6,7)(8,9)(10,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,11|d^2=e,c^a=cd,d^a=de,f^a=f^10,c^b=ce,d^c=de61 conjugacy classes — view the full character table.
| maximal subgroup of | #10622435, #10622440, #10622754, … |
| maximal normal subgroup of | #10622435, #10622440, #10622754, … |
| maximal quotient of | #10622608, #10622614, #10622616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 144);