C₃₁×#1422 #16841355

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃₁×#1422

Subgroups

	subgroup	G/H
center	C₁₆×C₃₁	D₄
commutator	C₄	C₂×C₁₆×C₃₁
Frattini, cosocle	C₂×C₁₆	C₂²×C₃₁
socle	C₂×C₃₁	C₄⋊C₁₆
prime radical	C₂²×C₃₁	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₃₁ ◁ C₂×C₁₆×C₃₁ ◁ G

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 2 generators on 63 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)(32,33,36,42,37,43,51,58,38,41,52,56,48,57,61,63,39,44,53,59,46,40,54,55,35,49,50,62,34,45,47,60);(32,34,35,46,39,48,38,37)(33,40,41,45,44,43,49,57)(36,47,50,54,53,61,52,51)(42,55,56,60,59,58,62,63)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7381, C₃₁×#7382, C₃₁×#7447, …
maximal normal subgroup of	C₃₁×#7381, C₃₁×#7382, C₃₁×#7447, …
maximal quotient of	C₃₁×#7455, C₃₁×#7458, C₃₁×#7467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 154);

C31×#1422 #16841355