Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | D4×C27 ⋊ C2 |
| Sylow | C27 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6171 |
| commutator, cosocle | C4×C27 | C22 |
| Frattini | C4×C9 | C2×D3 |
| Fitting | D4×C27 | C2 |
| socle | C2×C3 | #611 |
| hypercenter | D4 | D27 |
| hypocenter | C27 | D8 |
| LCS | G ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,8,23,9,27,13,11,25,5,4,19,6,21,16,24,22,7,14,12,2,15,10,20,26,3,17,18)(29,32)(30,35)(33,34);(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27)(28,29)(30,32)(31,34)(33,35)Polycyclic presentation:
2^4,3^3|c^2=d,e^3=f^2g,f^3=g^2,b^a=bc,c^a=cd,e^a=e^2f,f^a=f^2g,g^a=g^2,c^b=cd72 conjugacy classes — view the full character table.
| maximal subgroup of | D8×D27, #11720160, #11720163, … |
| maximal normal subgroup of | D8×D27, #11720160, #11720163, … |
| maximal quotient of | #11720047, #11720050, #11720066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 16);