#340×C₇₁ #17985471

order	4544
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#340×C₇₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	D₄
commutator	C₄	C₄²×C₇₁
Frattini, cosocle	C₂²×C₄	C₂²×C₇₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 2 generators on 87 points:

(1,2)(3,4)(5,6,7,8)(10,14)(11,12)(15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(2,4)(9,10,12,15,13,16,11,14)

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₁×#1347, C₇₁×#1348, C₂×#340×C₇₁, …
maximal normal subgroup of	C₇₁×#1347, C₇₁×#1348, C₂×#340×C₇₁, …
maximal quotient of	C₇₁×#1277, C₇₁×#1279, C₇₁×#1286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 21);

#340×C71 #17985471