Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of SU(3,𝔽2) |
| automorphism | Aut(G) = #10596418 (order 576) · Inn(G) = PSU(3,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | PSU(3,𝔽2) |
| commutator | #272 | C2×C4 |
| Fitting | C2×H(3,𝔽3) | Q8 |
| prime radical, cosocle | C2×#272 | C22 |
| hypocenter | H(3,𝔽3) | C4⋊C4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,17,7,8,15,5,19,24,4,26,6,12)(2,21,9)(3,23,10,16,25,11,27,20,14,13,22,18)(28,29,30,31);(2,5,6,16)(3,10,11,8)(7,20,21,26)(9,17,19,18)(12,14,22,23)(13,24,25,27)Polycyclic presentation:
2^4,3^3|a^2=c,b^2=cd,b^a=bc,e^a=ef^2,f^a=e^2f^2g^2,e^b=f,f^b=e^2g,e^c=e^2g,f^c=f^2g,f^e=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #6249⋊C4, (#1088⋊C2)⋊C4, #11722317, … |
| maximal normal subgroup of | #6249⋊C4, (#1088⋊C2)⋊C4, #11722317, … |
| maximal quotient of | #11720708, #11720709, #11720710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 239);