C₁₉×#12730 #18018689

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12730

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₁₉×#519 ◁ G

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

normal subgroups only

This group has 666 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)(20,21,25,30,26,31,22,32)(23,35,38,46,37,28,33,44)(24,29,34,42,40,45,39,27)(36,49,47,41,51,43,50,48);(20,22,26,25)(21,27,31,42)(23,33,37,38)(24,39,40,34)(28,41,35,48)(29,30,45,32)(36,50,51,47)(43,44,49,46);(20,23,26,37)(21,28)(22,33,25,38)(24,36,40,51)(27,41)(29,43)(30,44)(31,35)(32,46)(34,47,39,50)(42,48)(45,49)

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, 5718);