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C₄×C₉ #150

order	36
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃ (order 12) · Inn(G) = C₁ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = D₄×AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₆
abelian	C₄×C₉
multiplicative	(ℤ/37)ˣ, (ℤ/74)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₂×C₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

center of	C₉×M₁₆, C₉×#41, C₉×C₃⋊C₈, …
derived subgroup of	Dic₃₆, #3664, D₇₂, …
Frattini subgroup of	C₂₇⋊C₈, C₈×C₂₇, Dic₁₀₈, …
Fitting subgroup of	C₉⋊C₈, Dic₁₈, C₄×D₉, …
maximal subgroup of	C₉⋊C₈, C₈×C₉, Dic₁₈, …
maximal normal subgroup of	C₉⋊C₈, C₈×C₉, Dic₁₈, …
maximal quotient of	C₈×C₉, C₂×C₄×C₉, C₄×C₂₇, …
automorphism group of	C₃₇, C₂×C₃₇
outer automorphism group of	C₃₇, C₂×C₃₇, D₇₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 2);