C₁₉×C₃₇⋊C₃ #14523720

order	2109
isoclinism	isoclinic class of C₃₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×C₃₇⋊C₃
unambiguous semidirect product	C₃₇⋊(C₃×C₁₉)

Splittings

Fitting, socle, Hall	C₁₉×C₃₇ ⋊ C₃
derived, Sylow	C₃₇ ⋊ C₃×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	C₃₇⋊C₃
commutator, cosocle, hypocenter	C₃₇	C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₁₉×C₃₇	C₃
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 703

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22,24)(23,26,31)(25,29,28)(27,33,40)(30,35,37)(32,38,34)(36,43,47)(39,45,44)(41,48,42)(46,51,52)(49,53,50)(54,56,55);(20,21,23,27,33,41,49,54,53,38,24,28,34,42,43,26,32,39,46,48,31,37,44,50,51,35,22,25,30,36,40,47,52,55,56,45,29)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₉×C₃₇⋊C₃, C₁₉×D₃₇⋊C₃, D₁₉×C₃₇⋊C₃, …
maximal normal subgroup of	C₂×C₁₉×C₃₇⋊C₃, C₁₉×D₃₇⋊C₃, D₁₉×C₃₇⋊C₃, …
maximal quotient of	C₂×C₁₉×C₃₇⋊C₃, C₇×C₁₉×C₃₇⋊C₃, C₁₉×#11715913, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2109, 2);

C19×C37⋊C3 #14523720