C₁₁²×D₁₉ #17987151

order	4598
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁² · Schur multiplier (H₂) = C₁₁

Aliases

direct product	C₁₁²×D₁₉
unambiguous semidirect product	C₁₉⋊(C₂×C₁₁²)

Splittings

Fitting, socle, Hall	C₁₁²×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₁₁²

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁²	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂×C₁₁²
Frattini	C₁	G
Fitting, socle	C₁₁²×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₁₁²

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 2299

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5,6,7,8,9,10,11)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(12,13,14,15,16,17,18,19,20,21,22)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Conjugacy classes & character table

1331 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₁²×D₁₉⋊C₃, D₁₉⋊(C₁₁²⋊C₃), D₁₉×C₁₁²⋊C₃, …
maximal normal subgroup of	C₁₁²×D₁₉⋊C₃, D₁₉⋊(C₁₁²⋊C₃), D₁₉×C₁₁²⋊C₃, …
maximal quotient of	C₁₁²×D₅₇, C₃×C₁₁²×D₁₉, C₅×C₁₁²×D₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4598, 6);

C112×D19 #17987151