C₂²×C₈²×C₆₁ #29520743

order	15616
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

abelian

C₂²×C₈²×C₆₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle, prime radical	C₂⁴×C₆₁	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

1342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 15616

A minimal faithful permutation representation — 4 generators on 81 points:

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

Conjugacy classes & character table

15616 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₈×C₁₆×C₆₁, C₂×C₄×C₈²×C₆₁, C₂³×C₈²×C₆₁
maximal normal subgroup of	C₂²×C₈×C₁₆×C₆₁, C₂×C₄×C₈²×C₆₁, C₂³×C₈²×C₆₁
maximal quotient of	C₂²×C₈×C₁₆×C₆₁, C₂×C₄×C₈²×C₆₁, C₂³×C₈²×C₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 10298);

C22×C82×C61 #29520743