C₂×C₃₁×#504 #16843339

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₃₁×#504

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂²
commutator	C₂	C₂³×C₈×C₃₁
Frattini, cosocle	C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₈
prime radical	C₂×#41×C₃₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 65 points:

(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(3,4,6,10,7,11,16,22,8,12,17,23,18,24,28,32)(5,9,14,20,15,21,27,31,13,19,25,29,26,30,33,34);(3,5)(4,9)(6,14)(7,15)(8,13)(10,20)(11,21)(12,19)(16,27)(17,25)(18,26)(22,31)(23,29)(24,30)(28,33)(32,34);(5,13)(9,19)(14,25)(15,26)(20,29)(21,30)(27,33)(31,34)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12613, C₃₁×#12614, C₃₁×#12615, …
maximal normal subgroup of	C₃₁×#12613, C₃₁×#12614, C₃₁×#12615, …
maximal quotient of	C₂×C₃₁×#2108, C₄×C₃₁×#504, C₂×C₃₁×#2113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2138);

C2×C31×#504 #16843339