Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6196 |
| automorphism | Aut(G) = C2×#11722315 (order 1728) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C23 |
| Frattini, socle | C2×C3 | C2×D32 |
| Fitting | C4×H(3,𝔽3) | C22 |
| hypocenter | H(3,𝔽3) | #41 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C4 | |
156 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,9)(3,6,4)(5,8,7)(10,12,13,16)(11,15,14,17);(2,4)(3,7)(5,8)(6,9)(11,14)(15,17);(3,7)(5,6)(8,9)(10,11)(12,15)(13,14)(16,17)Polycyclic presentation:
2^4,3^3|c^2=d,b^a=bd,f^a=f^2,g^a=g^2,e^b=e^2,g^b=g^2,f^e=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #11720694, #11720840, #11720841, … |
| maximal normal subgroup of | #11720694, #11720840, #11720841, … |
| maximal quotient of | #11721091, #11721092, #11721096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 297);