C₁₉×#482 #11926158

order	1216
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#482

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₉
prime radical	C₁₉×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

106 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 608

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)(29,32)(30,33)(34,35);(21,24)(25,27)(28,29,31,32)(30,33,34,35);(20,22,23,26)(21,25,24,27)(28,30,31,34)(29,33,32,35)

Conjugacy classes & character table

361 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₉×#482, #14573547, #14573563, …
maximal subgroup of	C₂×C₁₉×#482, C₁₉×#3091, C₁₉×#3093, …
maximal normal subgroup of	C₂×C₁₉×#482, C₁₉×#3091, C₁₉×#3093, …
maximal quotient of	C₁₉×#1825, C₁₉×#1854, C₁₉×#1875, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 163);