C₁₉×#1391 #14559248

order	2432
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#1391

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#99
commutator	C₂³	C₄²×C₁₉
Frattini, cosocle	C₂×#41	C₂²×C₁₉
socle	C₂×C₁₉	#342
prime radical	C₂×#41×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×#41×C₁₉ ◁ G

166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,30)(23,34,35,27)(24,28)(25,29)(26,31,38,44)(32,42)(33,43)(36,47,48,40)(37,41)(39,45,49,51)(46,50);(20,22,24,32,25,33,37,46)(21,26,28,39,29,38,41,49)(23,31,36,45,35,44,48,51)(27,30,40,42,34,43,47,50)

Conjugacy classes & character table

494 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₉×#7398, C₁₉×#7399, C₁₉×#7406, …
maximal normal subgroup of	C₁₉×#7398, C₁₉×#7399, C₁₉×#7406, …
maximal quotient of	C₁₉×#7188, C₁₉×#7204, C₁₉×#7214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2432, 123);