C₂₉×#1696 #16815410

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂₉×#1696

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

140 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 61
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 3 generators on 61 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)(30,31,34,37)(32,38,40,45)(33,42,39,36)(35,43,44,41)(46,47,50,53)(48,54,56,61)(49,57,59,52)(51,58,60,55);(30,32,39,43,34,40,33,41)(31,35,42,45,37,44,36,38);(32,40)(35,44)(38,45)(41,43)(46,48,50,56)(47,51,53,60)(49,58,59,55)(52,61,57,54)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12388, C₂₉×#12389, C₂₉×#12412, …
maximal normal subgroup of	C₂₉×#12388, C₂₉×#12389, C₂₉×#12412, …
maximal quotient of	C₂₉×#8653, C₂₉×#9224, C₂₉×#9576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 428);