C₁₉×#464 #11926140

order	1216
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#464

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₉
prime radical	C₂×D₄×C₁₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ G

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 39
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25)(26,32)(27,29)(28,36)(30,34)(31,35)(33,39)(37,38);(20,21,22,23)(25,29)(30,38)(31,36)(32,39);(24,26,27,33)(25,30,29,38)(28,37,35,34)(31,39,36,32)

Conjugacy classes & character table

361 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₉×#464, #14573089, #14573105, …
maximal subgroup of	C₂×C₁₉×#464, C₁₉×#3035, C₁₉×#3037, …
maximal normal subgroup of	C₂×C₁₉×#464, C₁₉×#3035, C₁₉×#3037, …
maximal quotient of	C₁₉×#1678, C₁₉×#1680, C₁₉×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 145);