Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #11724027 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| derived | (C22⋊A4)⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | (C22⋊A4)⋊C9 | C2 |
| Frattini | C3 | A4⋊S4 |
| Fitting, socle | C24×C3 | C3⋊D3 |
| LCS | G ▷ (C22⋊A4)⋊C9 | |
| UCS | C1 | |
164 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3^3,2^4|b^3=d^2,c^3=d^2,b^a=b^2d,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,g^b=gh,h^b=g,e^c=ef,f^c=e30 conjugacy classes — view the full character table.
| maximal subgroup of | #11724023⋊C2, C2×#11724027, #72213207, … |
| maximal normal subgroup of | #11724023⋊C2, C2×#11724027, #72213207, … |
| maximal quotient of | #13164067, #13164071, #13164074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3994);