C₁₉×#535 #11926211

order	1216
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₉×#535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 31
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14564163, #14564168, #14564183, …
maximal subgroup of	C₁₉×#1910, C₁₉×#1911, C₁₉×#1914, …
maximal normal subgroup of	C₁₉×#1910, C₁₉×#1911, C₁₉×#1914, …
maximal quotient of	C₁₉×#2309, C₁₉×#2311, C₁₉×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 216);