C₂×C₂₉×#519 #16817140

order	3712
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×C₂₉×#519

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₉
Frattini, cosocle	C₂²	C₂⁵×C₂₉
prime radical	C₂⁴×C₂₉	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

1488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 5 generators on 51 points:

(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(3,4,5,6)(8,12)(10,18)(13,15)(16,22);(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)

Conjugacy classes & character table

1972 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8323, C₂₉×#10741, C₂₉×#10750, …
maximal normal subgroup of	C₂₉×#8323, C₂₉×#10741, C₂₉×#10750, …
maximal quotient of	C₂×C₂₉×#2274, C₂×C₂₉×#2276, C₄×C₂₉×#519, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2158);

C2×C29×#519 #16817140