C₁₉×#8188 #18014147

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#8188

Subgroups

	subgroup	G/H
center	C₄²×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

412 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 2432

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

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, 1176);