C₉×#2205 #11766926

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2205

Subgroups

	subgroup	G/H
center	C₂×C₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×Q₈	C₂³×C₃
socle	C₂×C₃	C₃×B(4,𝔽₂)
prime radical	C₃×2₋¹⁺⁴	C₂²×C₃
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₉ ◁ C₂×Q₈×C₉ ◁ G

336 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 84 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)(10,11)(12,20)(13,23)(14,18)(15,25)(16,21)(17,22)(19,24);(10,12,16,15)(11,17,21,23)(13,24,22,14)(18,20,19,25);(10,13,16,22)(11,18,21,19)(12,14,15,24)(17,25,23,20)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 937);