C₂×C₁₉×#1317 #18017383

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₁₉×#1317

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	#99
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₉
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂⁴×C₁₉ ◁ C₂²×C₂²⋊C₄×C₁₉ ◁ G

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

normal subgroups only

This group has 726 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) = 45
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(3,4,5,7,6,8,9,10)(11,12)(13,19)(14,22,23,17)(15,18)(16,20,24,26)(21,25);(11,13,15,21)(12,16,18,24)(14,20,23,26)(17,19,22,25)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 4412);