C₁₉×#11428 #18017387

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#11428

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#99
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

526 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 142 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)(20,21,22,24,23,25,26,27)(28,29)(30,38)(31,35)(32,45,46,36)(33,37)(34,40,48,55)(39,52)(41,53)(42,56)(43,58,59,50)(44,51)(47,54,49,57);(28,30,33,41)(29,34,37,48)(31,42,44,39)(32,40,46,55)(35,49,51,47)(36,38,45,53)(43,57,59,54)(50,56,58,52);(28,31,33,44)(29,35,37,51)(30,39,41,42)(32,43,46,59)(34,47,48,49)(36,50,45,58)(38,52,53,56)(40,54,55,57)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 4416);