Finite Group Databasebeta

← Prev (#152) Next (#154) →

C₂²×C₉ #153

order	36
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×D₃ (order 36) · Inn(G) = C₁ · Out(G) = C₂×C₃×D₃
holomorph	G ⋊ Aut(G) = S₄×AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

abelian	C₂²×C₉
multiplicative	(ℤ/57)ˣ, (ℤ/76)ˣ, (ℤ/108)ˣ, (ℤ/114)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃
socle, prime radical	C₂²×C₃	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2)(5,6,7,8,9,10,11,12,13);(3,4)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

center of	C₉×C₂²⋊C₄, C₉×C₄⋊C₄, C₂×D₄×C₉, …
derived subgroup of	D₉⋊A₄, A₄×D₉, #63365, …
Frattini subgroup of	C₄²⋊C₂₇, C₄×Dic₂₇, Dic₂₇⋊C₄, …
Fitting subgroup of	C₂×Dic₉, #611, C₂²×D₉, …
maximal subgroup of	C₂×Dic₉, #611, C₂×C₄×C₉, …
maximal normal subgroup of	C₂×Dic₉, #611, C₂×C₄×C₉, …
maximal quotient of	C₂×C₄×C₉, D₄×C₉, Q₈×C₉, …
automorphism group of	C₃×C₁₉, C₄×C₁₉, C₄×C₂₇, …
outer automorphism group of	C₃×C₁₉, C₄×C₁₉, C₄×C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 5);