C₂₉×#1698 #16815412

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

Aliases

direct product

C₂₉×#1698

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

156 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 53
transitive degree	μ_t(G) = 1856

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

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1698, C₂₉×#19155, C₂₉×#19156, …
maximal normal subgroup of	C₂×C₂₉×#1698, C₂₉×#19155, C₂₉×#19156, …
maximal quotient of	C₂₉×#8655, C₂₉×#9224, C₂₉×#9230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 430);