C₂×C₃₁×#431 #16842850

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₃₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×C₃₁×#431

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂²
commutator	C₂	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 49 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4)(5,8)(6,7)(9,10)(11,12,13,15,14,16,17,18);(4,7)(8,10);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

2480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7848, C₃₁×#7849, C₃₁×#7850, …
maximal normal subgroup of	C₃₁×#7848, C₃₁×#7849, C₃₁×#7850, …
maximal quotient of	C₂×C₃₁×#1744, C₂×C₄×C₃₁×C₂²⋊C₈, C₂×C₄×C₃₁×C₄⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1649);

C2×C31×#431 #16842850