Finite Group DatabasebetaFinite Group Databasebeta| order | 176 |
| automorphism | Aut(G) = C2×S4×AGL(1,𝔽11) (order 5280) · Inn(G) = C22×D11 · Out(G) = C2×C5×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C11 ⋊ C2 |
| Sylow | C11 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D11 |
| commutator, socle, cosocle | C2×C11 | C23 |
| Fitting | Q8×C11 | C2 |
| hypercenter | Q8 | D11 |
| hypocenter | C11 | #41 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 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)(12,14,15,18)(13,16,17,19);(12,13,15,17)(14,19,18,16);(2,3)(4,5)(6,7)(8,9)(10,11)(14,18)(16,19)Polycyclic presentation:
2^4,11|b^2=d,c^2=d,c^a=cd,e^a=e^10,c^b=cd35 conjugacy classes — view the full character table.
| maximal subgroup of | #66967, #66969, #66971, … |
| maximal normal subgroup of | #66967, #66969, #66971, … |
| maximal quotient of | #66943, #66945, #66946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(176, 34);