C₆₁×#13562 #29516995

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#13562

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₆₁
Frattini, cosocle	#127	C₂³×C₆₁
prime radical	C₆₁×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#532 ◁ G

490 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 976

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

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6550);