C₁₉×#1525 #14559382

order	2432
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#1525

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×C₂²⋊C₄
prime radical	C₁₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

270 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 35
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₉×#7486, C₁₉×#7492, C₁₉×#8241, …
maximal normal subgroup of	C₁₉×#7486, C₁₉×#7492, C₁₉×#8241, …
maximal quotient of	C₁₉×#7563, C₁₉×#7569, C₁₉×#7571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2432, 257);