Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4179 |
| automorphism | Aut(G) = (#4178⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #732 ⋊ C2 |
| derived | C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C9 | C2×C3 |
| Frattini | C32 | C3×D3 |
| Fitting | #732 | C2 |
| socle | C3 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,12,15,5,6,16)(3,21,23,9,10,22)(4,11)(7,18,25,8,19,26)(13,20,24,17,14,27);(1,20,5,11,8,7,4,2,14)(3,27,13,23,18,16,10,6,25)(9,26,12,21,15,19,22,17,24)Polycyclic presentation:
2,3^4|c^3=e^2,c^a=c^2e,d^a=d^2,e^a=e^2,c^b=cd,d^b=de13 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊C2, C2×#4179, #6878⋊C2, … |
| maximal normal subgroup of | #4178⋊C2, C2×#4179, C3×#4179, … |
| maximal quotient of | #66326, C2×#4179, #92160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 15);