Finite Group Databasebeta

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C₁₉×Dic₂₁ #13097599

order	1596
isoclinism	isoclinic class of D₂₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×Dic₂₁
unambiguous semidirect product	C₃⋊(C₁₉×Dic₇)
unambiguous semidirect product	C₇⋊(Dic₃×C₁₉)

Splittings

derived, Hall	C₃×C₇ ⋊ C₄×C₁₉
Hall	C₃×C₇×C₁₉ ⋊ C₄
Hall	C₇×C₁₉ ⋊ Dic₃
Hall	C₃×C₁₉ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₃×C₁₉
Sylow	C₃ ⋊ C₁₉×Dic₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	D₂₁
commutator, hypocenter	C₃×C₇	C₄×C₁₉
Frattini	C₂	C₁₉×D₂₁
Fitting, socle, prime radical	C₂×C₃×C₇×C₁₉	C₂
cosocle	C₂×C₃×C₇	C₂×C₁₉
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 1596

A minimal faithful permutation representation — 2 generators on 33 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);(20,22,21,23)(25,26)(28,33)(29,32)(30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₉×Dic₆₃, C₁₉×C₃⋊(ΓL(1,𝔽₈)⋊C₄), C₃⋊#13097586, …
maximal normal subgroup of	C₁₉×C₃⋊(ΓL(1,𝔽₈)⋊C₄), C₃⋊#13097586, C₃⋊#13097587, …
maximal quotient of	C₁₉×Dic₆₃, C₃×C₁₉×Dic₂₁, C₁₉×C₃⋊Dic₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1596, 22);