C₂₉×#1778 #16815492

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

Aliases

direct product

C₂₉×#1778

Subgroups

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

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1856

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#12651, C₂₉×#12678, C₂₉×#13114, …
maximal normal subgroup of	C₂₉×#12651, C₂₉×#12678, C₂₉×#13114, …
maximal quotient of	C₂₉×#8275, C₂₉×#8276, C₂₉×#8277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 510);

C29×#1778 #16815492