C₄×C₂₉×#125 #16816019

order	3712
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×C₂₉×#125

Subgroups

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

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 3712

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);(7,11)(9,12)(13,14,16,18)(15,17,19,20);(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,17,18,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#7568, C₂₉×#7585, C₂₉×#7602, …
maximal normal subgroup of	C₂₉×#7568, C₂₉×#7585, C₂₉×#7602, …
maximal quotient of	C₄×C₂₉×#387, C₄×C₂₉×#389, C₄×C₂₉×#382, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1037);

C4×C29×#125 #16816019