C₂₉×#1653 #16815367

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

Aliases

direct product

C₂₉×#1653

Subgroups

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

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

normal subgroups only

This group has 194 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) = 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)(31,34)(32,36)(35,37)(38,39,43,47)(40,48,51,52)(41,44,42,46)(45,49,53,50);(30,31,33,35)(32,37,36,34)(39,44)(41,42)(45,52)(46,47)(48,53)(49,50);(38,40,43,51)(39,45,47,53)(41,50,42,49)(44,52,46,48)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1653, C₂₉×#18503, C₂₉×#18518, …
maximal normal subgroup of	C₂×C₂₉×#1653, C₂₉×#18503, C₂₉×#18518, …
maximal quotient of	C₂₉×#7760, C₂₉×#7761, C₂₉×#7767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 385);