C₂₉×#2025 #16815739

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2025

Subgroups

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

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 49
transitive degree	μ_t(G) = 1856

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7428, C₂×C₂₉×#2025, C₂₉×#23451, …
maximal normal subgroup of	C₂₉×#7428, C₂×C₂₉×#2025, C₂₉×#23451, …
maximal quotient of	C₂₉×#10344, C₂₉×#10346, C₂₉×#10351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 757);