C₁₉×#12709 #18018668

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×#12709

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
prime radical	C₁₉×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

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

normal subgroups only

This group has 986 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)(20,21,24,28,25,29,40,47)(22,30,33,48,32,44,38,50)(23,34,37,46,36,27,39,45)(26,41,43,35,42,49,51,31);(20,22)(21,26)(23,35)(24,38)(25,32)(27,44)(28,43)(29,42)(30,34)(31,36)(33,40)(37,41)(39,49)(45,50)(46,48)(47,51);(23,36)(27,34)(30,44)(31,35)(37,39)(41,49)(45,46)(48,50)

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, 5697);