C₁₉×#12676 #18018635

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#12676

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#107
prime radical	C₂³×C₁₉	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 55
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25,27,32,40,47,54,55,29,31,42,46,28,33,41,49)(26,34,35,44,36,51,52,43,38,50,53,45,37,48,39,30);(24,26,28,36,29,38,40,37)(25,30,33,44,31,43,47,45)(27,39,41,35,42,52,54,53)(32,48,49,34,46,51,55,50);(20,22,23,21)(25,31)(30,43)(32,46)(33,47)(34,50)(44,45)(48,51)(49,55)

Conjugacy classes & character table

1216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5664);