#11797115

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ C₂×#365

Subgroups

	subgroup	G/H
center	C₂	C₂×AGL(1,𝔽₉)⋊C₄
commutator	#622	C₂²×C₄
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃⋊D₃	SD₁₆
cosocle	#3787	C₂³
hypercenter	C₂×C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	C₂×#365
LCS	G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

186 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^7,3^2|a^2=e,b^2=d,d^2=f,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=hi^2,i^a=h^2i^2,e^b=eg,h^b=h^2i,i^b=h,h^c=h^2,i^c=i^2,h^d=h^2i^2,i^d=h^2i,h^e=h^2,i^e=i^2,h^f=h^2,i^f=i^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252846
maximal normal subgroup of	#72252846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 31126);