C₁₉×#12172 #18018131

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#12172

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₁₉×#2190	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

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

normal subgroups only

This group has 344 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 — 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)(22,31)(23,26)(24,39)(25,30)(27,33)(28,35)(29,37)(32,45)(34,44)(36,40)(38,41)(42,51)(43,50)(46,48)(47,49);(21,26)(23,35)(24,37)(27,40)(28,29)(30,41)(31,45)(32,48)(34,49)(36,50)(38,51)(39,42)(43,44)(46,47);(20,22)(21,27,30,31)(23,36)(24,34)(25,33)(26,40,41,45)(28,43,42,46)(29,44,39,47)(32,38)(35,50)(37,49)(48,51)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5160);