Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6258 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C22⋊H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊S4 |
| commutator, hypocenter | C22⋊H(3,𝔽3) | C4 |
| Fitting | C23×C32 | D3 |
| socle | C23×C3 | C3⋊D3 |
| prime radical, cosocle | C2×C22⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4)(5,6,8,7)(10,12)(13,17)(14,16);(10,13,14)(12,16,17);(6,7,8)(9,10,12)(11,14,16)(13,17,15)Polycyclic presentation:
2^2,3^3,2^2|a^2=b,c^a=c^2,d^a=d^2,f^a=g,g^a=f,d^c=de,f^c=g,g^c=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊A4)⋊C4, ((C3⋊S4)⋊C3)⋊C4, #11722272, … |
| maximal normal subgroup of | ((C3⋊D3)⋊A4)⋊C4, ((C3⋊S4)⋊C3)⋊C4, #11722272, … |
| maximal quotient of | #11720736, #11720738, #11720740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 260);