C₃₇×AGL(1,ℤ/9) #14522007

order	1998
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×AGL(1,ℤ/9)
unambiguous semidirect product	D₉⋊(C₃×C₃₇)
unambiguous semidirect product	3₋¹⁺²⋊(C₂×C₃₇)

Splittings

Fitting, Hall	3₋¹⁺²×C₃₇ ⋊ C₂
derived	C₉ ⋊ C₂×C₃×C₃₇
Sylow	3₋¹⁺² ⋊ C₂×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₉	C₂×C₃×C₃₇
Frattini	C₃	C₃×D₃×C₃₇
Fitting	3₋¹⁺²×C₃₇	C₂
socle	C₃×C₃₇	C₃×D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₃₇

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 333

A minimal faithful permutation representation — 2 generators on 46 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)(39,43,46,41,42,45)(40,44);(38,46,41,44,42,43,40,39,45)

Conjugacy classes & character table

370 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1998, 37);

C37×AGL(1,ℤ/9) #14522007