C₁₉×#11425 #18017384

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#11425

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#99
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

678 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 43
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 43 points:

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

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, 4413);