Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ C2×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88954 |
| commutator, Fitting, cosocle | C33×C4 | C23 |
| Frattini, hypercenter | C4 | D33 |
| socle | C2×C33 | C2×D4 |
| prime radical | C2×C3⋊(C3⋊D3) | D4 |
| hypocenter | C33 | C2×Q16 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
354 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bd,c^a=cd,d^a=de,h^a=h^2,c^b=cd,d^b=de,g^b=g^2,f^c=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #13165351, #13165353, #13165366, … |
| maximal normal subgroup of | #13165351, #13165353, #13165366, … |
| maximal quotient of | #13155366, #13155472, #13155653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2815);