C₂₉×#1719 #16815433

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₂₉×#1719

Subgroups

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

186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 72 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)(47,52)(51,53);(30,31,35,37,33,36,32,34)(39,43)(44,51)(45,50)(46,52);(31,36)(34,37)(38,40,42,48,41,47,49,53)(39,44,45,46,43,51,50,52)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1719, C₂₉×#19613, C₂₉×#19619, …
maximal normal subgroup of	C₂×C₂₉×#1719, C₂₉×#19613, C₂₉×#19619, …
maximal quotient of	C₂₉×#8821, C₂₉×#9826, C₂₉×#9829, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 451);