Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #11921058 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C3×H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | C3×#120⋊C3 | C22 |
| Frattini | C23 | C2×C3⋊S4 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | GL(2,ℤ/4) |
| hypocenter | C3×C22⋊A4 | D4 |
| LCS | G ▷ C3×#120⋊C3 ▷ C3×C22⋊A4 | |
| UCS | C1 | |
298 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^3,3^2,2^4|b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,f^b=fhi,g^b=gh,f^c=fh,g^c=gi,f^d=g,g^d=fg,h^d=i,i^d=hi33 conjugacy classes — view the full character table.
| maximal subgroup of | #72219823, #72241554, D3×#4625⋊C2, … |
| maximal normal subgroup of | D3×#4625⋊C2, C2×#11921058 |
| maximal quotient of | #72258621, C2×#11921058 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155069);