Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620184 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | #1113⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620184 |
| commutator, hypocenter | #1113⋊C3 | C4 |
| Fitting, socle | C2×C33 | S4 |
| prime radical, cosocle | C2×#1113⋊C3 | C2 |
| LCS | G ▷ #1113⋊C3 | |
| UCS | C1 ◁ C2 | |
112 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 9 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:
2^2,3,2^2,3^3|a^2=b,c^a=c^2,d^a=e,e^a=d,f^a=g^2,g^a=f^2,h^a=h^2,d^c=e,e^c=de,f^c=h,g^c=f,h^c=g,f^d=f^2,h^d=h^2,g^e=g^2,h^e=h^228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3083);