C₅×C₁₁×Dic₁₉ #16862011

order	4180
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₁×Dic₁₉
unambiguous semidirect product	C₁₉⋊(C₄×C₅×C₁₁)

Splittings

derived, Sylow	C₁₉ ⋊ C₄×C₅×C₁₁
Hall	C₅×C₁₁×C₁₉ ⋊ C₄
Hall	C₁₁×C₁₉ ⋊ C₄×C₅
Hall	C₅×C₁₉ ⋊ C₄×C₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₁₁	D₁₉
commutator, hypocenter	C₁₉	C₄×C₅×C₁₁
Frattini	C₂	C₅×C₁₁×D₁₉
Fitting, socle, prime radical	C₂×C₅×C₁₁×C₁₉	C₂
cosocle	C₂×C₁₉	C₂×C₅×C₁₁
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅×C₁₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 4180

A minimal faithful permutation representation — 2 generators on 39 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);(17,19,18,20)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

1210 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅×C₁₁×Dic₁₉, C₅×C₁₁×Dic₅₇, C₅×C₁₁×(C₁₉⋊C₃)⋊C₄, …
maximal normal subgroup of	C₃×C₅×C₁₁×Dic₁₉, C₅×C₁₁×(C₁₉⋊C₃)⋊C₄, C₁₁×C₂₅×Dic₁₉, …
maximal quotient of	C₃×C₅×C₁₁×Dic₁₉, C₅×C₁₁×Dic₅₇, C₁₁×C₂₅×Dic₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4180, 3);

C5×C11×Dic19 #16862011