Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88610 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊S4 |
| commutator, prime radical, cosocle, hypocenter | Q8⋊H(3,𝔽3) | C2 |
| Fitting | C32×Q8 | D3 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,15,10,7,5,6,12,13)(2,3,14,9,8,11,16,4)(18,20)(21,25)(22,24);(18,21,22)(20,24,25);(3,9,12)(4,10,11)(6,13,16)(7,14,15)(17,18,20)(19,22,24)(21,25,23)Polycyclic presentation:
2,3^3,2^3|a^2=g,e^2=g,f^2=g,b^a=b^2,c^a=c^2,e^a=f,f^a=e,c^b=cd,e^b=fg,f^b=ef,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #11722253, #11722255, #11722257, … |
| maximal normal subgroup of | #11722253, #11722255, #11722257, … |
| maximal quotient of | #11720738, C2×#88610, #13046987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 257);