Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of C3⋊D15 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(2,𝔽3) (order 17280) · Inn(G) = C3⋊D15 · Out(G) = C2×#985 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊D15 |
| unambiguous semidirect product | (C2×C3)⋊D15 |
| unambiguous semidirect product | (C2×C5)⋊(C3⋊D3) |
| Fitting, socle | C2×C32×C5 ⋊ C2 |
| derived, Hall | C32×C5 ⋊ C22 |
| Sylow | C32 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D15 |
| commutator, cosocle, hypocenter | C32×C5 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(4,5)(7,8)(10,11)(12,13);(1,2)(3,4,5)(9,10,12,13,11);(6,7,8)48 conjugacy classes — view the full character table.
| maximal subgroup of | #67137, #67139, #67140, … |
| maximal normal subgroup of | #67137, #67139, #67140, … |
| maximal quotient of | #67180, C4×C3⋊D15, C4⋊(C3⋊D15), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 36);