#11771027

order	1152
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1292

Subgroups

	subgroup	G/H
center	C₂	#10593166
commutator	C₂×#157	C₄²
Frattini, hypercenter	C₂×C₄	AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₂×C₄×#157	C₂²
hypocenter	C₃²	#1292
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

186 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 72

Polycyclic presentation:

2^7,3^2|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=fg,b^a=bc,c^a=cf,e^a=efg,h^a=hi,i^a=i^2,d^b=dg,h^b=h^2i,i^b=h,h^c=h^2i^2,i^c=h^2i,h^e=h^2i^2,i^e=h^2i,h^f=h^2,i^f=i^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220480
maximal normal subgroup of	#72220480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 5038);