C₁₉×#12224 #18018183

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#12224

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₉
prime radical	C₂³×C₁₉	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂×C₄⋊C₄×C₁₉ ◁ G

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

normal subgroups only

This group has 380 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) = 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)(20,21,25,31)(22,34,37,47)(23,39,41,50)(24,43,35,29)(26,33,46,32)(27,42,49,36)(28,45,48,40)(30,44,51,38);(20,22,25,37)(21,27,31,49)(23,40,41,45)(24,42,35,36)(26,38,46,44)(28,51,48,30)(29,47,43,34)(32,39,33,50);(20,23,26,42)(21,28,33,29)(22,35,38,40)(24,44,45,37)(25,41,46,36)(27,47,50,51)(30,49,34,39)(31,48,32,43)

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, 5212);