C₁₉×#8827 #18014786

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#8827

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#384
prime radical	C₂³×C₁₉	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

306 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 51
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27,25,28,23,33)(22,29,32,34,30,35,31,26)(36,37)(38,44)(39,41)(40,48)(42,46)(43,47)(45,50)(49,51);(20,22,23,31,25,30,24,32)(21,26,27,29,28,34,33,35)(37,41)(42,49)(43,48)(44,50);(22,30)(26,34)(29,35)(31,32)(36,38,40,46,39,45,47,51)(37,42,43,44,41,49,48,50)

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, 1815);