C₁₉×#12724 #18018683

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12724

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 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)(22,31)(23,35,37,28)(24,29)(25,30)(26,32,42,49)(27,36,43,51)(33,46)(34,47)(38,40)(39,44)(41,48,45,50);(20,22,25,34)(21,26,30,42)(23,32,37,49)(24,38,39,33)(27,40,43,46)(28,31,35,47)(29,45,44,41)(36,50,51,48);(20,23,25,37)(21,27)(22,32,34,49)(24,36,39,51)(26,40)(28,44)(29,35)(30,43)(31,41)(33,48,38,50)(42,46)(45,47)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5712);