Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊(C3⋊S4) |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽3)×H(3,𝔽3) | C2 |
| Fitting | Q8×H(3,𝔽3) | D3 |
| LCS | G ▷ SL(2,𝔽3)×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
278 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=d^2,f^a=fg,g^a=gh,f^b=g,g^b=fgh,d^c=de,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #72242719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2134);