C₁₉×#551 #11926227

order	1216
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#551

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
prime radical	C₁₉×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 304

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

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14565625, D₁₉×#551, #14577190, …
maximal subgroup of	C₁₉×#2197, C₂×C₁₉×#551, C₁₉×#3475, …
maximal normal subgroup of	C₁₉×#2197, C₂×C₁₉×#551, C₁₉×#3475, …
maximal quotient of	C₁₉×#2342, C₁₉×#2343, C₁₉×#2353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 232);