C₂₉×#1837 #16815551

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1837

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₉	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 53 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)(32,35)(33,34)(36,37)(38,39,42,48,44,49,43,47)(40,46,41,45,51,52,50,53);(31,34)(35,37)(38,40,43,50,44,51,42,41)(39,45,48,52,49,53,47,46);(30,32,33,36)(31,35,34,37)(38,41,42,51,44,50,43,40)(39,46,47,53,49,52,48,45)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7219, C₂₉×#7220, C₂₉×#7223, …
maximal normal subgroup of	C₂₉×#7219, C₂₉×#7220, C₂₉×#7223, …
maximal quotient of	C₂₉×#8784, C₂₉×#8785, C₂₉×#8789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 569);

C29×#1837 #16815551