C₁₉×#397 #11926073

order	1216
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#397

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

170 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 70 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)(24,25,26,27)(29,32)(33,34);(21,23)(28,29)(30,34)(31,32)(33,35);(20,21)(22,23)(24,25)(26,27)(28,30,31,35)(29,33,32,34)

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14561635, #14562620, #14562625, …
maximal subgroup of	C₁₉×#1344, C₁₉×#1607, C₁₉×#1608, …
maximal normal subgroup of	C₁₉×#1344, C₁₉×#1607, C₁₉×#1608, …
maximal quotient of	C₁₉×#1439, C₁₉×#1440, C₁₉×#1442, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 78);