C₄×C₇×C₃₁×C₃₇ #44545786

order	32116
nilpotency class	1
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₂₁₁₆
abelian	C₄×C₇×C₃₁×C₃₇
multiplicative	(ℤ/32117)ˣ, (ℤ/64234)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₇×C₃₁×C₃₇
socle, prime radical	C₂×C₇×C₃₁×C₃₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 32116

A minimal faithful permutation representation — 1 generator on 79 points:

(1,2,3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)

Conjugacy classes & character table

32116 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Dic₁₆₀₅₈, D₃₂₁₁₆
maximal subgroup of	Dic₁₆₀₅₈, D₃₂₁₁₆
maximal normal subgroup of	Dic₁₆₀₅₈, D₃₂₁₁₆
automorphism group of	C₃₂₁₁₇
outer automorphism group of	C₃₂₁₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32116, 1);

C4×C7×C31×C37 #44545786