Finite Group Databasebeta

← Prev (#162) Next (#164) →

C₃₇ #163

order	37
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₉ (order 36) · Inn(G) = C₁ · Out(G) = C₄×C₉
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₃₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₃₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₁

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 37

A minimal faithful permutation representation — 1 generator on 37 points:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₃₇, D₅×C₃₇, A₄×C₃₇, …
derived subgroup of	D₃₇, C₃₇⋊C₃, Dic₃₇, …
Frattini subgroup of	C₁₃₆₉, D₁₃₆₉, C₂×C₁₃₆₉, …
Fitting subgroup of	D₃₇, C₃₇⋊C₃, #3866, …
socle of	D₃₇, C₃₇⋊C₃, #3866, …
maximal subgroup of	D₃₇, C₂×C₃₇, C₃₇⋊C₃, …
maximal normal subgroup of	D₃₇, C₂×C₃₇, C₃₇⋊C₃, …
maximal quotient of	C₂×C₃₇, C₃×C₃₇, C₅×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37, 1);