Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×D4×C11 ⋊ C2 |
| Sylow | C11 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D11 |
| commutator, socle, cosocle | C2×C11 | C24 |
| Fitting | C2×D4×C11 | C2 |
| hypercenter | C2×D4 | D11 |
| hypocenter | C11 | H(5,𝔽2) |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 88 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,4,6,8,10,11,9,7,5,3)(13,17)(16,19);(14,18)(16,19);(12,14)(13,16)(15,18)(17,19);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^5,11|d^a=de,f^a=f^10,c^b=ce67 conjugacy classes — view the full character table.
| maximal subgroup of | #10622354, #10622355, #10622359, … |
| maximal normal subgroup of | #10622354, #10622355, #10622359, … |
| maximal quotient of | #10623014, #10623017, #10623021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 179);