Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620185 |
| 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 | #10620185 |
| commutator, hypocenter | #1113⋊C3 | C4 |
| Fitting, socle | C2×C33 | S4 |
| prime radical, cosocle | C2×#1113⋊C3 | C2 |
| LCS | G ▷ #1113⋊C3 | |
| UCS | C1 ◁ C2 | |
104 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 104 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.
| minimal degree | μ(G) = 13 |
| 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=h,g^a=fgh^2,h^a=f,d^c=e,e^c=de,f^c=h,g^c=f^2g^2h,h^c=g^2h,g^d=g^2,h^d=h^2,f^e=f^2,g^e=g^2h^234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3086);