Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #157⋊D3 |
| automorphism | Aut(G) = C22×D3×GO+(4,𝔽2) (order 1728) · Inn(G) = #157⋊D3 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #157⋊D3 |
| commutator | C3×C3⋊D3 | C2×C4 |
| Fitting, socle | C2×C33 | D4 |
| prime radical | C2×#1113 | C2 |
| cosocle | C2×C3×C3⋊D3 | C22 |
| hypocenter | C33 | C22⋊C4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
132 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3)(4,8,6,5)(7,9)(10,11,12,13);(2,3)(7,8)Polycyclic presentation:
2^4,3^3|b^2=cd,b^a=bc,e^a=f,f^a=e,g^a=g^2,e^b=f^2,f^b=e,e^c=e^2,f^c=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D32⋊C4, (D32⋊D3)⋊C4, #11724112, … |
| maximal normal subgroup of | D3×D32⋊C4, (D32⋊D3)⋊C4, #11724112, … |
| maximal quotient of | #11722780, #11722781, #11722782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 586);