Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×Q8×C11 ⋊ C2 |
| Sylow | C11 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D11 |
| commutator, socle, cosocle | C22×C11 | C23 |
| Fitting | C2×Q8×C11 | C2 |
| prime radical | C2×D44 | C2 |
| hypercenter | C2×Q8 | D11 |
| hypocenter | C11 | #124 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,8,10,11,9,7,5,3)(13,15)(16,18,19,23)(17,21,20,22);(16,17,19,20)(18,22,23,21);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(17,20)(21,22)Polycyclic presentation:
2^5,11|b^2=e,c^2=e,b^a=be,c^a=cd,f^a=f^10,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #10622115, #10622366, #10622408, … |
| maximal normal subgroup of | #10622115, #10622366, #10622408, … |
| maximal quotient of | #10622590, #10622599, #10622601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 142);