C₂₉×#1890 #16815604

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

Aliases

direct product

C₂₉×#1890

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂⁴	C₂³×C₂₉
prime radical	C₂²×D₄×C₂₉	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂⁴×C₂₉ ◁ G

386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 49
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 49 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)(31,33)(34,35)(36,39)(37,44,45,40)(38,41)(42,46,48,49)(43,47);(30,31,32,33)(34,36,38,43)(35,39,41,47)(37,42,45,48)(40,46,44,49);(34,37)(35,40)(36,42)(38,45)(39,46)(41,44)(43,48)(47,49)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12754, C₂₉×#12760, C₂₉×#12761, …
maximal normal subgroup of	C₂₉×#12754, C₂₉×#12760, C₂₉×#12761, …
maximal quotient of	C₂₉×#8966, C₂₉×#8968, C₂₉×#9011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 622);