Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #88622⋊C3 |
| unambiguous semidirect product | #88722⋊C3 |
| unambiguous semidirect product | (C3⋊(C3⋊(#41⋊C3)))⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32×Q8 | C2×C32 |
| Frattini, socle | C2×C3 | C3×D3×A4 |
| Fitting | Q8×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 | |
191 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^2,3^2,2|a^2=h,d^2=h,e^2=h,f^a=f^2,g^a=g^2,d^b=de,e^b=d,f^c=fg,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #72242717 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2152);