C₄²×C₁₆×C₅₃ #27220049

order	13568
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄³

Aliases

abelian

C₄²×C₁₆×C₅₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(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);(17,18,19,20);(21,22,23,24)

Conjugacy classes & character table

13568 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄²×C₃₂×C₅₃, C₄×C₈×C₁₆×C₅₃, C₂×C₄²×C₁₆×C₅₃
maximal normal subgroup of	C₄²×C₃₂×C₅₃, C₄×C₈×C₁₆×C₅₃, C₂×C₄²×C₁₆×C₅₃
maximal quotient of	C₄²×C₃₂×C₅₃, C₄×C₈×C₁₆×C₅₃, C₂×C₄²×C₁₆×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5525);

C42×C16×C53 #27220049