C₂₉×#1910 #16815624

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#1910

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	#386
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂₉×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

344 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 45
transitive degree	μ_t(G) = 464

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

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12755, C₂₉×#12758, C₂₉×#12778, …
maximal normal subgroup of	C₂₉×#12755, C₂₉×#12758, C₂₉×#12778, …
maximal quotient of	C₂₉×#9101, C₂₉×#9102, C₂₉×#9105, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 642);