Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C22×C3⋊D3 |
| commutator, cosocle | C22×H(3,𝔽3) | C23 |
| Fitting | C24×H(3,𝔽3) | C2 |
| hypercenter | C24×C3 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | #120 |
| LCS | G ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 | |
715 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^5,3^3|b^a=be,c^a=cd,f^a=f^2,g^a=g^2,g^f=gh106 conjugacy classes — view the full character table.
| maximal subgroup of | #13136957, #13141177, #13144449, … |
| maximal normal subgroup of | #13136957, #13141177, #13144449, … |
| maximal quotient of | #13144808, #13144811, #13144825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1843);