#343×C₇₁ #17985474

order	4544
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#343×C₇₁

Subgroups

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

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 2272

A minimal faithful permutation representation — 2 generators on 83 points:

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

Conjugacy classes & character table

1562 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₁×#1345, C₇₁×#1346, C₇₁×#1353, …
maximal normal subgroup of	C₇₁×#1345, C₇₁×#1346, C₇₁×#1353, …
maximal quotient of	C₇₁×#1280, C₇₁×#1283, C₇₁×#1285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 24);

#343×C71 #17985474