C₁₉×#7188 #18013147

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

Aliases

direct product

C₁₉×#7188

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂³	C₄×C₈×C₁₉
Frattini, cosocle	C₂×#117	C₂²×C₁₉
socle	C₂²×C₁₉	#342
prime radical	C₂⁴×C₁₉	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 59
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 59 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)(30,38)(31,42,43,35)(32,36)(33,37)(34,39,46,52)(40,50)(41,51)(44,55,56,48)(45,49)(47,53,57,59)(54,58);(28,30,32,40,33,41,45,54)(29,34,36,47,37,46,49,57)(31,39,44,53,43,52,56,59)(35,38,48,50,42,51,55,58)

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, 176);