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C₂×C₉² #4187

order	162
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×#13049364 (order 3888) · Inn(G) = C₁ · Out(G) = C₃×#13049364
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₉

Aliases

C₂×C₉²

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃²	C₂×C₃²
socle, prime radical	C₂×C₃²	C₃²
LCS	G ▷ C₁
UCS	C₁ ◁ G

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(3,4,5,6,7,8,9,10,11);(12,13,14,15,16,17,18,19,20)

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₉², Q₈×C₉², C₉²×Dic₃, …
derived subgroup of	#10619529, #10619530, #10619531, …
Fitting subgroup of	C₉×Dic₉, C₉⋊Dic₉, C₂×C₉×D₉, …
maximal subgroup of	C₉×Dic₉, C₉⋊Dic₉, C₄×C₉², …
maximal normal subgroup of	C₉×Dic₉, C₉⋊Dic₉, C₄×C₉², …
maximal quotient of	C₄×C₉², C₂²×C₉², C₂×#6855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 23);