C₁₉×#577 #11926253

order	1216
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#577

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ #41×C₁₉ ◁ G

258 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 158 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)(20,21,24,28)(22,26,29,33)(23,25,31,27)(30,32,35,34);(21,25)(23,31)(26,32)(27,28)(30,35)(33,34);(20,22)(21,26)(23,30)(24,29)(25,32)(27,34)(28,33)(31,35);(22,29)(26,33)(30,35)(32,34)

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14575397, #14575408, #14575415, …
maximal subgroup of	C₁₉×#3288, C₁₉×#3289, C₁₉×#3290, …
maximal normal subgroup of	C₁₉×#3288, C₁₉×#3289, C₁₉×#3290, …
maximal quotient of	C₁₉×#2897, C₁₉×#2898, C₁₉×#2912, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 258);