C₁₉×#7312 #18013271

order	4864
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#7312

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1304
commutator	C₂⁴	C₄²×C₁₉
Frattini, cosocle	C₂²×#41	C₂²×C₁₉
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

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

normal subgroups only

This group has 456 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 — 2 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)(22,30,32,41)(23,34,35,27)(24,37,38,28)(25,29)(26,31)(33,48)(36,42)(39,45,50,51)(40,46)(43,49)(44,47);(20,22,24,33,25,32,38,47)(21,26,28,39,29,40,37,50)(23,31,36,45,35,46,49,51)(27,41,42,48,34,30,43,44)

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, 300);