C₉×#2047 #11766768

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2047

Subgroups

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂³×C₃	C₃×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

384 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,19,23)(16,24,27,39)(17,28,31,21)(18,32,34,22)(20,33,36,26)(25,41,29,40)(30,38,45,43)(35,42,37,44);(14,16,19,27)(15,20,23,36)(17,29,31,25)(18,33,34,26)(21,37,28,35)(22,39,32,24)(30,42,45,44)(38,40,43,41);(11,13)(14,17,18,30,19,31,34,45)(15,21,22,38,23,28,32,43)(16,25,26,42,27,29,33,44)(20,35,24,40,36,37,39,41)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 779);