C₂₉×#1638 #16815352

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1638

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂×D₄×C₂₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 194 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) = 61
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 61 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)(30,31,35,38)(32,39,41,44)(33,37,34,42)(36,43,45,40)(46,47,51,55)(48,56,59,60)(49,54,50,53)(52,58,61,57);(30,32)(31,36)(33,40)(34,43)(35,41)(37,44)(38,45)(39,42)(46,48,51,59)(47,52,55,61)(49,58,50,57)(53,60,54,56);(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)(47,53)(49,50)(52,60)(54,55)(56,61)(57,58)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1638, C₂₉×#18494, C₂₉×#18521, …
maximal normal subgroup of	C₂×C₂₉×#1638, C₂₉×#18494, C₂₉×#18521, …
maximal quotient of	C₂₉×#7765, C₂₉×#7767, C₂₉×#7773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 370);