C₉×AGL(1,ℤ/16) #11766902

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂×C₉	#442
commutator	C₈	C₂²×C₄×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#442
prime radical	C₃×#509	C₂×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#97 ◁ G

231 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,4,5,6,7,8,9)(11,15,17,24)(12,19)(13,20)(16,22,25,18)(21,23);(10,11)(12,16)(13,22)(14,17)(15,23)(18,20)(19,24)(21,25);(10,12,20,23,14,21,13,19)(11,16,18,15,17,25,22,24)

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₉×AGL(1,ℤ/16)
maximal subgroup of	D₉×AGL(1,ℤ/16), 3₋¹⁺²×AGL(1,ℤ/16)
maximal normal subgroup of	D₉×AGL(1,ℤ/16), 3₋¹⁺²×AGL(1,ℤ/16)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 913);