C₃₇⋊C₃ #1126

order	111
isoclinism	isoclinic class of C₃₇⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₃₇) (order 1332) · Inn(G) = G · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃₇⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₃₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₇	C₃
LCS	G ▷ C₃₇
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 37 points:

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

Polycyclic presentation:

3,37|b^a=b^26

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₇⋊C₃, C₂×C₃₇⋊C₃, #66526, …
maximal normal subgroup of	D₃₇⋊C₃, C₂×C₃₇⋊C₃, #66526, …
maximal quotient of	C₂×C₃₇⋊C₃, #66524, C₃×C₃₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(111, 1);

C37⋊C3 #1126