C₁₉×#12779 #18018738

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12779

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1790
commutator	C₂×Q₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×#41×C₁₉ ◁ G

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

normal subgroups only

This group has 732 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)(22,32,35,46)(23,29,24,37)(25,30)(26,31)(27,33)(28,36,45,39)(34,48,51,41)(38,43)(40,44)(42,49)(47,50);(20,22,26,35)(21,27,31,43)(23,33,24,38)(25,34,40,51)(28,41,45,48)(29,46,37,32)(30,47,44,42)(36,49,39,50);(20,23)(21,28)(22,33)(24,26)(25,39)(27,41)(29,44)(30,37)(31,45)(32,47)(34,50)(35,38)(36,40)(42,46)(43,48)(49,51)

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, 5767);