C₂₉×#1617 #16815331

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1617

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 202 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) = 928

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

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12221, C₂₉×#12222, C₂₉×#12223, …
maximal normal subgroup of	C₂₉×#12221, C₂₉×#12222, C₂₉×#12223, …
maximal quotient of	C₂₉×#7792, C₂₉×#7794, C₂₉×#7806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 349);