Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C27 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6171 |
| commutator, cosocle | C4×C27 | C22 |
| Frattini | C4×C9 | C2×D3 |
| Fitting | Q8×C27 | C2 |
| socle, prime radical | C2×C3 | #611 |
| hypercenter | Q8 | D27 |
| hypocenter | C27 | Q16 |
| LCS | G ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 43 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)(28,30,32,38)(29,33,35,42)(31,40,41,37)(34,43,39,36);(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)(28,29,32,35)(30,36,38,43)(31,39,41,34)(33,40,42,37)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=d,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 | #11720164, #11720165, Q16×D27, … |
| maximal normal subgroup of | #11720164, #11720165, Q16×D27, … |
| maximal quotient of | #11720047, #11720049, #11720076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 17);