#11776398

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1549

Subgroups

	subgroup	G/H
center	C₂	C₂×PSU(3,𝔽₂)⋊C₄
commutator	C₂×#157	C₂²×C₄
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	C₂×#103
prime radical	C₂²×C₃⋊D₃	C₂²⋊C₄
cosocle	C₄×#157	C₂³
hypercenter	C₂×C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#1549
LCS	G ▷ C₂×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

274 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 62 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=e,b^2=fg,d^2=g,e^2=fg,b^a=bd,c^a=cf,d^a=df,h^a=h^2i,i^a=h,d^b=dg,e^b=efg,h^b=i^2,i^b=h,h^d=hi,i^d=hi^2,h^e=h^2i^2,i^e=h^2i,h^g=h^2,i^g=i^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220197, #72220226, #72220240
maximal normal subgroup of	#72220197, #72220226, #72220240

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 10409);