C₁₉×#12277 #18018236

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12277

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

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 1216

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)(20,21,25,28)(22,29,31,34)(23,27,24,32)(26,33,35,30)(36,37)(38,42)(39,49)(40,53)(41,45)(43,52)(44,54)(46,61)(47,60)(48,57)(50,65)(51,64)(55,62)(56,63)(58,66)(59,67);(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32)(36,38,41,48)(37,42,45,57)(39,50,51,46)(40,47,54,63)(43,58,59,55)(44,60,53,56)(49,61,64,65)(52,66,67,62);(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)(36,39)(37,43,45,59)(38,46)(40,52)(41,51)(42,55,57,58)(44,49,53,64)(47,62)(48,50)(54,67)(56,61,60,65)(63,66)

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, 5265);