C₉×#1819 #11766540

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉×#1819

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁵×C₉ ◁ G

756 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 168 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)(12,18,20,16)(13,15)(14,17)(19,24,25,22)(21,23);(11,15)(16,22)(17,23)(18,24);(10,12,13,19)(11,16,15,22)(14,20,21,25)(17,18,23,24)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 551);