Finite Group DatabasebetaFinite Group Databasebeta| order | 176 |
| isoclinism | isoclinic class of #4323 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽11) (order 1760) · Inn(G) = C22⋊D11 · Out(G) = C22×C5 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C11 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D11 |
| commutator, cosocle | C4×C11 | C22 |
| Frattini | C4 | C2×D11 |
| Fitting | Q8×C11 | C2 |
| socle, prime radical | C2×C11 | D4 |
| hypercenter | Q8 | D11 |
| hypocenter | C11 | Q16 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,4,6,8,10,11,9,7,5,3)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^4,11|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^10,c^b=cd32 conjugacy classes — view the full character table.
| maximal subgroup of | #66968, #66969, Q16×D11, … |
| maximal normal subgroup of | #66968, #66969, Q16×D11, … |
| maximal quotient of | #66871, #66874, #66899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(176, 17);