C₃₁×#3459 #16843392

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

Aliases

direct product

C₃₁×#3459

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 5 generators on 47 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)(34,38)(35,37)(36,39)(40,41)(42,45)(43,44)(46,47);(32,34)(33,36)(35,39)(37,38);(34,39)(36,38)(41,44)(45,47);(33,37)(36,38);(40,42,43,46)(41,45,44,47)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8317, C₃₁×#8399, C₃₁×#8401, …
maximal normal subgroup of	C₃₁×#8317, C₃₁×#8399, C₃₁×#8401, …
maximal quotient of	C₃₁×#34057, C₃₁×#34082, C₃₁×#34083, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2191);

C31×#3459 #16843392