C₁₉×#12710 #18018669

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12710

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
prime radical	C₁₉×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂³×Q₈×C₁₉ ◁ G

952 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 146 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)(20,21,25,30,26,31,23,28)(22,32,33,50,35,47,41,49)(24,37,36,48,40,29,39,46)(27,42,45,38,44,51,43,34);(20,22)(21,27)(23,33)(24,38)(25,41)(26,35)(28,43)(29,47)(30,45)(31,44)(32,37)(34,40)(36,42)(39,51)(46,49)(48,50);(20,23,26,25)(21,28,31,30)(22,33,35,41)(24,39,40,36)(27,43,44,45)(29,48,37,46)(32,49,47,50)(34,42,38,51)

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, 5698);