C₃₁×#2352 #16842285

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₁×#2352

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 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)(34,35)(36,37,38,39)(41,42)(45,46);(33,35)(40,41,44,42)(43,46,47,45);(40,42,44,41)(43,46,47,45);(32,33)(34,35)(40,43)(41,45)(42,46)(44,47)

Conjugacy classes & character table

1550 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#22414, C₃₁×#22416, C₃₁×#22420, …
maximal normal subgroup of	C₃₁×#22414, C₃₁×#22416, C₃₁×#22420, …
maximal quotient of	C₂²⋊C₄×C₄⋊C₄×C₃₁, C₃₁×#13801, C₃₁×#13804, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1084);

C31×#2352 #16842285