C₁₉×#7315 #18013274

order	4864
nilpotency class	4
isoclinism	isoclinic class of #7313
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#7315

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1304
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂³×Q₈	C₂²×C₁₉
prime radical	C₂⁴×C₁₉	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂³×Q₈×C₁₉ ◁ G

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

normal subgroups only

This group has 432 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)(20,21,24,29,26,31,39,47)(22,32,34,48,35,42,50,51)(23,30,37,46,25,28,38,44)(27,40,41,33,43,49,45,36);(20,22,25,33,26,35,23,36)(21,27,30,42,31,43,28,32)(24,34,37,40,39,50,38,49)(29,45,44,51,47,41,46,48)

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, 303);