C₉×#1415 #11766136

order	1152
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1415

Subgroups

	subgroup	G/H
center	C₂²×C₉	D₁₆
commutator	C₁₆	C₂×C₄×C₉
Frattini, cosocle	C₂×C₃×C₁₆	C₂²×C₃
socle	C₂²×C₃	C₃×D₁₆
prime radical	C₂×C₃×D₁₆	C₂×C₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₈×C₉ ◁ C₂×C₉×C₁₆ ◁ G

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

normal subgroups only

This group has 123 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 147);