C₂₉×#1884 #16815598

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1884

Subgroups

	subgroup	G/H
center	C₄×C₂₉	#120
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂×C₂₉	#386
prime radical	C₂₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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)(32,39,40,43)(33,36,41,42)(34,37)(35,38)(44,45);(30,32,34,42,35,40,44,36)(31,33,37,43,38,41,45,39);(30,33)(31,36)(32,37)(34,43)(35,41)(38,42)(39,44)(40,45)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7482, C₂₉×#7484, C₂₉×#7486, …
maximal normal subgroup of	C₂₉×#7482, C₂₉×#7484, C₂₉×#7486, …
maximal quotient of	C₂₉×#8957, C₂₉×#8958, C₂₉×#8959, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 616);