C₁₉×#539 #11926215

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

Aliases

direct product

C₁₉×#539

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
prime radical	C₁₉×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 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)(22,23)(24,25)(26,27)(30,34)(32,35);(21,25)(22,26)(28,29,31,33)(30,32,34,35);(20,22,24,26)(21,23,25,27)(28,30)(29,32)(31,34)(33,35);(20,23)(21,26)(22,25)(24,27)

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14562564, #14562598, #14562632, …
maximal subgroup of	C₁₉×#1597, C₁₉×#1603, C₁₉×#1609, …
maximal normal subgroup of	C₁₉×#1597, C₁₉×#1603, C₁₉×#1609, …
maximal quotient of	C₁₉×#2313, C₁₉×#2318, C₁₉×#2322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 220);