Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #95 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator | C2×H(3,𝔽3) | C42 |
| Frattini, socle | C23×C3 | D32 |
| Fitting, prime radical, cosocle | C23×H(3,𝔽3) | C22 |
| hypocenter | H(3,𝔽3) | #95 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
320 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=c,b^a=be,g^a=g^2,h^a=h^2,f^b=f^2,h^b=h^2,g^f=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #13136711, #13136712, #13140000, … |
| maximal normal subgroup of | #13136711, #13136712, #13140000, … |
| maximal quotient of | #13135158, #13135159, #13135162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 301);