#324×C₇₁ #17985455

order	4544
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#324×C₇₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²⋊C₄
commutator	C₂²	C₂×C₈×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

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 2272

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇₁×#1318, C₇₁×#1319, C₇₁×#1324, …
maximal normal subgroup of	C₇₁×#1318, C₇₁×#1319, C₇₁×#1324, …
maximal quotient of	C₇₁×#1271, C₇₁×#1280, C₇₁×#1322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 5);

#324×C71 #17985455