Finite Group DatabasebetaFinite Group Databasebeta| order | 176 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽11) (order 1760) · Inn(G) = D11 · Out(G) = C2×C5×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C11⋊C8 |
| unambiguous semidirect product | (C2×C11)⋊C8 |
| derived, Sylow | C11 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D11 |
| commutator, hypocenter | C11 | C2×C8 |
| Frattini | C4 | C2×D11 |
| Fitting | C2×C4×C11 | C2 |
| socle, prime radical | C22×C11 | C4 |
| cosocle | C4×C11 | C22 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C4 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5,7,6,8,9,10)(12,13)(14,15)(16,17)(18,19)(20,21);(1,2)(11,12,14,16,18,20,21,19,17,15,13)56 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C11⋊C8, #66867, C4⋊(C11⋊C8), … |
| maximal normal subgroup of | C4×C11⋊C8, #66867, C4⋊(C11⋊C8), … |
| maximal quotient of | C4×C11⋊C8, C4⋊(C11⋊C8), C2×C11⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(176, 8);