C₁₉×#12207 #18018166

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

Aliases

direct product

C₁₉×#12207

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂×C₂²⋊C₄×C₁₉ ◁ G

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

normal subgroups only

This group has 388 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) = 59
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 59 points:

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

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, 5195);