Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88390 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊C27 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C22⋊C27)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C27 | C2 |
| Frattini | C2×C9 | S4 |
| Fitting | Q8×C9 | D3 |
| socle | C2×C3 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C2 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,8,3,4,7,6,5)(10,13)(11,17)(12,20)(14,15)(16,26)(18,19)(21,28)(22,27)(23,33)(24,32)(25,31)(29,30)(34,35);(2,5,8)(3,6,7)(9,28,22,29,33,31,11,10,32,12,35,26,18,15,14,19,16,34,20,24,13,17,25,23,30,27,21)Polycyclic presentation:
2,3^3,2^3|b^3=c^2d,c^3=d^2,e^2=g,f^2=g,b^a=b^2c,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,e^b=fg,f^b=ef,f^e=fg36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#88391, #11720649, #11720651, … |
| maximal normal subgroup of | C2×#88391, #11720649, #11720651, … |
| maximal quotient of | #11720108, C2×#88391, #13046340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 38);