C₃₁×#3285 #16843218

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#3285

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 51 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)(34,35)(37,41)(38,44)(39,47)(42,45)(43,48)(50,51);(36,37,40,43)(38,45,44,50)(39,48,47,41)(42,49,51,46);(33,35);(36,38,40,44)(37,42,43,51)(39,49,47,46)(41,50,48,45)

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#33831, C₃₁×#33835, C₃₁×#33836, …
maximal normal subgroup of	C₃₁×#33831, C₃₁×#33835, C₃₁×#33836, …
maximal quotient of	C₃₁×#19835, C₃₁×#19836, C₃₁×#19837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2017);

C31×#3285 #16843218