C₃₇×D₁₉⋊C₃ #16862425

order	4218
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×D₁₉⋊C₃
unambiguous semidirect product	D₁₉⋊(C₃×C₃₇)
unambiguous semidirect product	(C₁₉⋊C₃)⋊(C₂×C₃₇)

Splittings

Fitting, socle, Hall	C₁₉×C₃₇ ⋊ C₂×C₃
derived, Sylow	C₁₉ ⋊ C₂×C₃×C₃₇
Hall	C₃₇×C₁₉⋊C₃ ⋊ C₂
Hall	C₃₇×D₁₉ ⋊ C₃
Hall	C₁₉⋊C₃ ⋊ C₂×C₃₇
Hall	D₁₉ ⋊ C₃×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	D₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂×C₃×C₃₇
Frattini	C₁	G
Fitting, socle	C₁₉×C₃₇	C₂×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃₇

16 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)(39,43,44,40,41,45)(42,51,48,46,47,53)(49,54,56,55,52,50);(38,39,42,48,54,51,55,41,45,50,56,44,43,49,47,52,53,46,40)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₃₇×D₁₉⋊C₃, C₃₇×C₅⋊(D₁₉⋊C₃), C₇×C₃₇×D₁₉⋊C₃, …
maximal normal subgroup of	C₅×C₃₇×D₁₉⋊C₃, C₇×C₃₇×D₁₉⋊C₃
maximal quotient of	C₅×C₃₇×D₁₉⋊C₃, C₃₇×C₅⋊(D₁₉⋊C₃), C₇×C₃₇×D₁₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4218, 18);

C37×D19⋊C3 #16862425