C₂₉×#1626 #16815340

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

Aliases

direct product

C₂₉×#1626

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 210 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) = 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)(31,35)(32,33)(36,37)(39,43)(40,44)(41,45);(30,31)(32,35)(33,37)(34,36)(38,39,42,40)(41,44,45,43);(30,32,34,33)(31,35,36,37)(38,40,42,39)(41,43,45,44)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1626, C₂₉×#18494, C₂₉×#18518, …
maximal normal subgroup of	C₂×C₂₉×#1626, C₂₉×#18494, C₂₉×#18518, …
maximal quotient of	C₂₉×#7739, C₂₉×#7743, C₂₉×#7747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 358);