C₂₉×#1687 #16815401

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1687

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 198 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) = 45
transitive degree	μ_t(G) = 1856

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)(31,34)(32,35)(36,37)(38,39)(40,42)(41,44)(43,45);(30,31,35,37,33,36,32,34);(31,36)(34,37)(39,42)(40,45)(43,44)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12324, C₂₉×#12325, C₂₉×#12348, …
maximal normal subgroup of	C₂₉×#12324, C₂₉×#12325, C₂₉×#12348, …
maximal quotient of	C₂₉×#8652, C₂₉×#9229, C₂₉×#9571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 419);