C₁₉×#7127 #18013086

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#7127

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#327
commutator	C₂×C₈	C₂×C₈×C₁₉
Frattini, cosocle	C₂×#97	C₂²×C₁₉
prime radical	C₂³×C₁₉	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

202 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 1216

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 115);