C₁₉×#7427 #18013386

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₁₉
Frattini, cosocle	#381	C₂²×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂⁴×C₁₉ ◁ C₁₉×#381 ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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)(20,21,24,28,25,29,38,45)(22,30,32,46,33,47,49,51)(23,34,36,50,37,27,42,43)(26,39,40,31,41,48,44,35);(20,22)(21,26)(23,35,37,31)(24,32)(25,33)(27,30,34,47)(28,44)(29,41)(36,48,42,39)(38,49)(40,45)(43,46,50,51)

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, 415);