C₉×#2199 #11766920

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₉×#2199

Subgroups

	subgroup	G/H
center	C₂×C₉	B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₉
Frattini, cosocle	C₂×C₃×D₄	C₂³×C₃
socle	C₂×C₃	C₃×B(4,𝔽₂)
prime radical	C₃×#2199	C₃
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₉ ◁ C₂×D₄×C₉ ◁ G

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

normal subgroups only

This group has 405 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)(12,18)(13,16)(14,17)(15,20)(19,24)(21,23)(22,25);(10,12)(11,15)(13,20)(14,19)(16,18)(17,22)(21,25)(23,24);(12,19)(13,21)(15,22)(18,24)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 931);