C₃₇×C₁₉⋊C₃ #14523719

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

Aliases

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

Splittings

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

Subgroups

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

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×C₁₉⋊C₃, C₁₉⋊C₃×D₃₇, C₃₇×D₁₉⋊C₃, …
maximal normal subgroup of	C₂×C₃₇×C₁₉⋊C₃, C₁₉⋊C₃×D₃₇, C₃₇×D₁₉⋊C₃, …
maximal quotient of	C₂×C₃₇×C₁₉⋊C₃, C₇×C₃₇×C₁₉⋊C₃, C₃₇×#87704, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2109, 1);

C37×C19⋊C3 #14523719