C₂₉×#2076 #16815790

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2076

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂⁴×C₂₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

256 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 49 points:

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7077, C₂₉×#7085, C₂₉×#12281, …
maximal normal subgroup of	C₂₉×#7077, C₂₉×#7085, C₂₉×#12281, …
maximal quotient of	C₂₉×#10616, C₂₉×#10618, C₂₉×#10622, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 808);