Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C32×D4 ⋊ D4 |
| Sylow | C32 ⋊ #3484 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C25 |
| Fitting | C2×C32×D4 | D4 |
| socle | C2×C32 | C23×D4 |
| hypercenter | C2×D4 | GO+(4,𝔽2) |
| hypocenter | C32 | #3484 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
3030 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^7,3^2|b^a=bf,c^a=cg,e^a=ef,i^a=hi^2,d^b=df,h^c=hi,i^c=i^2,h^g=h^2,i^g=i^281 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×GO+(4,𝔽2), #72241543, #72253196 |
| maximal normal subgroup of | H(5,𝔽2)×GO+(4,𝔽2), #72241543, #72253196 |
| automorphism group of | #3797, #3802 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 138373);