C₉×#1350 #11766071

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1350

Subgroups

	subgroup	G/H
center	C₂²×C₉	#102
commutator	C₂×C₈	C₂×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₈	C₂²×C₃
socle	C₂²×C₃	C₃×#102
prime radical	C₂³×C₃	C₃×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

156 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 49
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 2 generators on 49 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,16,17,14)(13,15)(18,19,22,27,24,29,23,28)(20,30,32,43,34,47,33,46)(21,35,37,26,39,44,38,45)(25,36,40,31,42,48,41,49);(10,12)(11,14)(13,17)(15,16)(18,20,24,34)(19,25,29,42)(21,36,39,48)(22,33,23,32)(26,43,45,46)(27,41,28,40)(30,35,47,44)(31,37,49,38)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 82);