C₁₉×#12297 #18018256

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12297

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₁₉×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

290 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 67
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27)(22,25,29,32)(23,26,30,33)(28,34,35,31)(36,37)(38,50)(39,53)(40,56)(41,48)(42,47)(43,49)(44,51)(45,55)(46,59)(52,64)(54,63)(57,66)(58,61)(60,65)(62,67);(20,22)(21,25)(23,28)(24,29)(26,31)(27,32)(30,35)(33,34)(36,38,43,39)(37,44,49,63)(40,57,60,67)(41,52,42,55)(45,61,64,46)(47,62,48,66)(50,65,53,56)(51,58,54,59);(20,23,24,30)(21,26,27,33)(22,28,29,35)(25,31,32,34)(36,39,43,38)(37,45,49,64)(40,54,60,51)(41,52,42,55)(44,61,63,46)(47,50,48,53)(56,62,65,66)(57,59,67,58)

Conjugacy classes & character table

589 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5285);