C₂₉×#1682 #16815396

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1682

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

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

normal subgroups only

This group has 166 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) = 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)(32,35)(36,37)(38,39)(40,46)(41,43)(42,50)(44,48)(45,49)(47,53)(51,52);(30,31,35,37,33,36,32,34)(39,43)(44,52)(45,50)(46,53);(38,40,41,47)(39,44,43,52)(42,51,49,48)(45,53,50,46)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1682, C₂₉×#19123, C₂₉×#19125, …
maximal normal subgroup of	C₂×C₂₉×#1682, C₂₉×#19123, C₂₉×#19125, …
maximal quotient of	C₂₉×#8649, C₂₉×#9218, C₂₉×#9222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 414);