C₂₉×#2122 #16815836

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#2122

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#99
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×D₄	C₂³×C₂₉
prime radical	C₂₉×#535	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂×C₄×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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)(32,36)(33,40,41,35)(34,37)(38,44)(39,43,45,42);(31,35)(32,38)(36,43)(37,40)(39,45)(42,44);(30,32,34,38)(31,36,37,44)(33,39,41,45)(35,42,40,43)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12715, C₂₉×#12718, C₂₉×#12721, …
maximal normal subgroup of	C₂₉×#12715, C₂₉×#12718, C₂₉×#12721, …
maximal quotient of	C₂₉×#11427, C₂₉×#11454, C₂₉×#11473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 854);