C₂×C₃₁×#578 #16843516

order	3968
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₁ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×C₃₁×#578

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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);(3,4,8,12)(5,13,16,26)(6,10,17,24)(7,19,21,11)(9,20,23,15)(14,27,28,34)(18,31,30,25)(22,32,33,29);(3,5,8,16)(4,9,12,23)(6,14,17,28)(7,20,21,15)(10,22,24,33)(11,26,19,13)(18,32,30,29)(25,34,31,27);(3,6)(4,10)(5,14)(7,18)(8,17)(9,22)(11,25)(12,24)(13,27)(15,29)(16,28)(19,31)(20,32)(21,30)(23,33)(26,34);(6,17)(10,24)(14,28)(18,30)(22,33)(25,31)(27,34)(29,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12742, C₃₁×#12744, C₃₁×#12749, …
maximal normal subgroup of	C₃₁×#12742, C₃₁×#12744, C₃₁×#12749, …
maximal quotient of	C₂×C₃₁×#2898, C₂×C₃₁×#2913, C₂×C₃₁×#2947, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2315);

C2×C31×#578 #16843516