C₁₉×#12217 #18018176

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#12217

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₉
prime radical	C₂⁵×C₁₉	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂×C₂²⋊C₄×C₁₉ ◁ G

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

normal subgroups only

This group has 484 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)(20,21,24,28)(22,30,33,41)(23,34,37,27)(25,29,38,43)(26,35,40,32)(31,44,46,51)(36,47,49,42)(39,48,50,45);(20,22,24,33)(21,26,28,40)(23,35,37,32)(25,31,38,46)(27,41,34,30)(29,39,43,50)(36,48,49,45)(42,51,47,44);(22,31)(23,36)(26,39)(30,44)(33,46)(37,49)(40,50)(41,51)

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, 5205);