C₄×C₂₉×#131 #16816588

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₄×C₂₉×#131

Subgroups

	subgroup	G/H
center	C₄×C₈×C₂₉	C₂²
commutator	C₂	C₂²×C₄²×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₉ ◁ G

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 49 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)

Conjugacy classes & character table

2320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8184, C₂₉×#8185, C₂₉×#8186, …
maximal normal subgroup of	C₂₉×#8184, C₂₉×#8185, C₂₉×#8186, …
maximal quotient of	C₈×C₂₉×#131, C₂₉×#17320, C₂₉×#17328, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1606);

C4×C29×#131 #16816588