C₂₉×#1805 #16815519

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1805

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

292 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 3712

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

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8429, C₂₉×#8431, C₂₉×#8433, …
maximal normal subgroup of	C₂₉×#8429, C₂₉×#8431, C₂₉×#8433, …
maximal quotient of	C₂₉×#8566, C₂₉×#8570, C₂₉×#8574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 537);