C₁₉×D₁₁₉ #17985288

order	4522
isoclinism	isoclinic class of D₁₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₁₁₉
unambiguous semidirect product	C₇⋊(C₁₉×D₁₇)
unambiguous semidirect product	C₁₇⋊(D₇×C₁₉)

Splittings

Fitting, socle, Hall	C₇×C₁₇×C₁₉ ⋊ C₂
derived, Hall	C₇×C₁₇ ⋊ C₂×C₁₉
Hall	C₁₇×C₁₉ ⋊ D₇
Hall	C₇×C₁₉ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₇×C₁₉
Sylow	C₇ ⋊ C₁₉×D₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₁₁₉
commutator, cosocle, hypocenter	C₇×C₁₇	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₁₉	C₂
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₁₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 2261

A minimal faithful permutation representation — 2 generators on 43 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(21,26)(22,25)(23,24)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

1159 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₁₉×D₁₁₉, C₁₉×D₃₅₇, C₁₉⋊C₃×D₁₁₉, …
maximal normal subgroup of	C₃×C₁₉×D₁₁₉, C₁₉⋊C₃×D₁₁₉, C₁₉×C₁₇⋊AGL(1,𝔽₇), …
maximal quotient of	C₃×C₁₉×D₁₁₉, C₁₉×D₃₅₇, C₅×C₁₉×D₁₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4522, 7);

C19×D119 #17985288