#323×C₇₁ #17985454

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

Aliases

direct product

#323×C₇₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²⋊C₄
commutator	C₂²	C₂×C₈×C₇₁
Frattini, cosocle	C₂²×C₄	C₂²×C₇₁
prime radical	C₂⁴×C₇₁	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) = 87
transitive degree	μ_t(G) = 1136

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

(1,2,3,7,5,8,10,14)(4,6,9,12,11,13,15,16)(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);(2,6)(3,9)(8,13)(10,15)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇₁×#1316, C₇₁×#1317, C₇₁×#1326, …
maximal normal subgroup of	C₇₁×#1316, C₇₁×#1317, C₇₁×#1326, …
maximal quotient of	C₇₁×#1270, C₇₁×#1272, C₇₁×#1280, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 4);

#323×C71 #17985454