C₁₉×#575 #11926251

order	1216
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#575

Subgroups

	subgroup	G/H
center	C₄×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 304

A minimal faithful permutation representation — 4 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)(22,29)(23,28)(24,32)(25,27)(26,30)(31,35)(33,34);(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(21,27)(26,34)(28,32)(29,35);(20,23,25,24)(21,28,27,32)(22,30,31,33)(26,35,34,29)

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14563999, #14564138, #14564685, …
maximal subgroup of	C₁₉×#1881, C₁₉×#1906, C₁₉×#2008, …
maximal normal subgroup of	C₁₉×#1881, C₁₉×#1906, C₁₉×#2008, …
maximal quotient of	C₁₉×#2896, C₁₉×#2911, C₁₉×#2943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 256);