C₁₉×#387 #11926063

order	1216
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#387

Subgroups

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

154 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 90 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)(21,23)(24,25,27,26)(29,32)(30,34);(24,26,27,25)(28,29)(30,33)(31,32)(34,35);(20,21)(22,23)(28,30,31,34)(29,33,32,35)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₉×#387, #14566073, #14566108, …
maximal subgroup of	C₂×C₁₉×#387, C₁₉×#2290, C₁₉×#2295, …
maximal normal subgroup of	C₂×C₁₉×#387, C₁₉×#2290, C₁₉×#2295, …
maximal quotient of	C₁₉×#1434, C₁₉×#1436, C₁₉×#1441, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 68);