Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C22×C3⋊D3 |
| commutator, cosocle | C2×H(3,𝔽3) | C23 |
| Fitting | Q8×H(3,𝔽3) | C2 |
| hypercenter | C3×Q8 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | #41 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,5,6)(4,8,9)(10,12,13,16)(11,14,15,17);(1,2,4)(3,6,8)(5,9,7)(10,11,13,15)(12,17,16,14);(2,4)(5,9)(6,8)(12,16)(14,17)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^2,c^b=cd,f^e=fg50 conjugacy classes — view the full character table.
| maximal subgroup of | #11720791, #11721227, #11721229, … |
| maximal normal subgroup of | #11720791, #11721227, #11721229, … |
| maximal quotient of | #11721758, #11721760, #11721761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 395);