Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #1088⋊C2 |
| automorphism | Aut(G) = #13164097 (order 1728) · Inn(G) = #1088⋊C2 · 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, hypercenter | C2 | #1088⋊C2 |
| commutator | #272 | C2×C4 |
| Frattini, socle | C2×C3 | GO+(4,𝔽2) |
| Fitting | C2×H(3,𝔽3) | D4 |
| prime radical, cosocle | C2×#272 | C22 |
| hypocenter | H(3,𝔽3) | C4⋊C4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,13,11,5,12)(2,6,15)(4,10)(7,9,8,16,18,14)(19,21,22,20);(1,2)(3,9)(4,7)(5,14)(6,10)(8,13)(11,18)(12,15)(16,17)(19,20,22,21)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=d,b^a=bc,e^a=e^2,g^a=g^2,e^b=f,f^b=e,g^b=g^2,e^c=e^2,f^c=f^2,f^e=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #6249⋊C4, SU(3,𝔽2)⋊C4, #11722300, … |
| maximal normal subgroup of | #6249⋊C4, SU(3,𝔽2)⋊C4, #11722300, … |
| maximal quotient of | #11720693, #11720697, #11720703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 237);