C₁₉×#12367 #18018326

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12367

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×C₈×C₁₉ ◁ G

336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 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)(22,30,32,49)(23,34)(24,37)(26,41,43,51)(27,45)(28,40)(31,36,33,48)(35,47)(38,39)(42,46,44,50);(20,21,25,29)(22,26,32,43)(23,35,39,45)(24,40,37,28)(27,34,47,38)(30,46,49,50)(31,44,33,42)(36,41,48,51);(20,22)(21,26)(23,36)(24,31)(25,32)(27,46)(28,42)(29,43)(30,38)(33,37)(34,49)(35,41)(39,48)(40,44)(45,51)(47,50)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5355);