C₂×C₄×C₃₂×C₅₃ #27221058

order	13568
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄×C₃₂×C₅₃
multiplicative	(ℤ/25455)ˣ, (ℤ/27152)ˣ, (ℤ/33940)ˣ, (ℤ/50910)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 91 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(33,34,35,36);(37,38)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₄×C₅₃×C₆₄, C₂×C₈×C₃₂×C₅₃, C₄²×C₃₂×C₅₃, …
maximal normal subgroup of	C₂×C₄×C₅₃×C₆₄, C₂×C₈×C₃₂×C₅₃, C₄²×C₃₂×C₅₃, …
maximal quotient of	C₂×C₄×C₅₃×C₆₄, C₂×C₈×C₃₂×C₅₃, C₄²×C₃₂×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6534);

C2×C4×C32×C53 #27221058