C₂₉×#1790 #16815504

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂₉×#1790

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂⁴	C₂³×C₂₉
prime radical	C₂×C₂₉×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂⁴×C₂₉ ◁ G

670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 41
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 41 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)(32,33)(34,35)(36,40,41,38)(37,39);(34,36)(35,38)(37,41)(39,40);(31,33)(34,35)(36,38)(37,39)(40,41)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#13041, C₂₉×#13042, C₂₉×#13043, …
maximal normal subgroup of	C₂₉×#13041, C₂₉×#13042, C₂₉×#13043, …
maximal quotient of	C₂₉×#8316, C₂₉×#8317, C₂₉×#8346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 522);