C₁₉×#12304 #18018263

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12304

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₁₉×#3259	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

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

normal subgroups only

This group has 286 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)(20,21,26,30,25,31,24,41)(22,32,42,49,34,28,35,45)(23,36,37,46,39,27,40,29)(33,43,50,44,38,47,51,48);(21,27)(23,37)(24,26)(28,43)(29,41)(30,46)(31,36)(32,47)(33,50)(35,42)(38,51)(39,40)(44,49)(45,48);(20,22,25,34)(21,28)(23,38,39,33)(24,35,26,42)(27,43)(29,44)(30,45)(31,32)(36,47)(37,51,40,50)(41,49)(46,48)

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, 5292);