C₂₉×#1764 #16815478

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#1764

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₉	D₄
commutator	C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

300 conjugacy classes of subgroups.

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)(30,31)(32,33)(34,35,38,42,39,43,37,40)(36,41,45,48,46,49,44,47);(31,33)(35,40,43,42)(41,47,49,48);(34,36,38,45,39,46,37,44)(35,41,42,48,43,49,40,47)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#12614, C₂₉×#12615, C₂₉×#12917, …
maximal normal subgroup of	C₂₉×#12614, C₂₉×#12615, C₂₉×#12917, …
maximal quotient of	C₂₉×#8184, C₂₉×#8186, C₂₉×#8187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 496);

C29×#1764 #16815478