C₂²×C₃₁×#125 #16843370

order	3968
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₁ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×C₃₁×#125

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3968

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

(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(3,4);(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

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8306, C₃₁×#8335, C₃₁×#8554, …
maximal normal subgroup of	C₃₁×#8306, C₃₁×#8335, C₃₁×#8554, …
maximal quotient of	C₂²×C₃₁×#382, C₂²×C₃₁×#387, C₂²×C₃₁×#389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2169);

C22×C31×#125 #16843370