C₁₉×#2117 #14559974

order	2432
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#2117

Subgroups

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

220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(20,21,24,30,25,31,39,48,26,28,40,44,23,29,37,46)(22,32,34,49,33,42,50,47,35,45,38,51,36,27,41,43);(20,22)(21,27)(23,36)(24,38)(25,33)(26,35)(28,42)(29,45)(30,47)(31,32)(34,40)(37,50)(39,41)(43,44)(46,49)(48,51);(21,28)(27,42)(29,31)(30,44)(32,45)(43,47)(46,48)(49,51)

Conjugacy classes & character table

836 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₉×#12609, C₁₉×#12618, C₁₉×#12685, …
maximal normal subgroup of	C₁₉×#12609, C₁₉×#12618, C₁₉×#12685, …
maximal quotient of	C₁₉×#11421, C₁₉×#11422, C₁₉×#11443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2432, 849);