C₁₉×#2135 #14559992

order	2432
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#2135

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×Q₈	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×#41×C₁₉ ◁ G

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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)(20,21,25,30,26,31,23,28)(22,32,35,50,36,47,33,49)(24,38,37,48,40,29,41,46)(27,42,43,39,44,51,45,34);(20,22,26,36)(21,27,31,44)(23,35,25,33)(24,39,40,34)(28,45,30,43)(29,47,38,32)(37,42,41,51)(46,49,48,50);(20,23,26,25)(21,28,31,30)(22,33,36,35)(24,37,40,41)(27,43,44,45)(29,46,38,48)(32,49,47,50)(34,42,39,51)

Conjugacy classes & character table

437 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₉×#12712, C₁₉×#12713, C₁₉×#12721, …
maximal normal subgroup of	C₁₉×#12712, C₁₉×#12713, C₁₉×#12721, …
maximal quotient of	C₁₉×#11434, C₁₉×#11450, C₁₉×#11481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2432, 867);