C₄×C₁₉×#355 #18018588

order	4864
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×C₁₉×#355

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂×C₄	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#99
prime radical	C₂²×D₄×C₁₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

532 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 39
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(5,6,8,12,9,13,17,20)(7,11,15,19,14,10,16,18);(6,10)(7,14)(11,13)(12,19)(15,16)(18,20)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5617);