C₁₉×#7125 #18013084

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

Aliases

direct product

C₁₉×#7125

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#327
commutator	C₂×C₈	C₂×C₈×C₁₉
Frattini, cosocle	C₂×#97	C₂²×C₁₉
socle	C₂×C₁₉	#1271
prime radical	C₂³×C₁₉	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

182 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 608

A minimal faithful permutation representation — 2 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)(22,30,32,43,33,47,38,50)(23,34,36,27)(24,29)(25,28)(26,31,40,48,41,49,42,35)(37,46,51,44)(39,45);(20,22)(21,26,28,41)(23,35)(24,38)(25,33)(27,43,34,50)(29,40,45,42)(30,46,47,44)(31,37)(32,39)(36,48)(49,51)

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, 113);