C₁₉×#7147 #18013106

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#7147

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#95
commutator	C₂×C₄	C₄×C₈×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

188 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 67
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,29,31,35)(23,27)(24,33)(25,28)(26,30,34,32)(36,37,39,43,40,45,50,57,41,44,51,56,52,58,63,66)(38,46,48,59,49,60,62,67,47,42,53,54,61,55,64,65);(20,22,24,32,25,31,23,30)(21,26,27,29,28,34,33,35)(36,38)(37,42,44,46)(39,48)(40,49)(41,47)(43,54,56,59)(45,55,58,60)(50,62)(51,53)(52,61)(57,65,66,67)(63,64)

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, 135);