Finite Group Databasebeta

← Prev (#16811815) Next (#16811817) →

C₅×C₁₉×Dic₉ #16811816

order	3420
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₉×Dic₉
unambiguous semidirect product	C₉⋊(C₄×C₅×C₁₉)

Splittings

derived, Sylow	C₉ ⋊ C₄×C₅×C₁₉
Hall	C₅×C₉×C₁₉ ⋊ C₄
Hall	C₉×C₁₉ ⋊ C₄×C₅
Hall	C₅×C₉ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₁₉	D₉
commutator, hypocenter	C₉	C₄×C₅×C₁₉
Frattini	C₂×C₃	C₅×D₃×C₁₉
Fitting	C₂×C₅×C₉×C₁₉	C₂
socle, prime radical	C₂×C₃×C₅×C₁₉	D₃
cosocle	C₂×C₉	C₂×C₅×C₁₉
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₅×C₁₉

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 3420

A minimal faithful permutation representation — 2 generators on 37 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);(25,27,26,28)(30,37)(31,36)(32,35)(33,34)

Conjugacy classes & character table

1140 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₉×C₂₅×Dic₉, C₅²×C₁₉×Dic₉, C₅×C₁₉×Dic₄₅, …
maximal normal subgroup of	C₁₉×C₂₅×Dic₉, C₅²×C₁₉×Dic₉, C₅×C₇×C₁₉×Dic₉, …
maximal quotient of	C₁₉×C₂₅×Dic₉, C₅²×C₁₉×Dic₉, C₅×C₁₉×Dic₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3420, 9);