C₆₁×#7519 #29510952

order	15616
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#7519

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₆₁
Frattini, cosocle	#492	C₂²×C₆₁
prime radical	C₆₁×#1624	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₄²×C₆₁ ◁ C₆₁×#492 ◁ G

162 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 77 points:

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

Conjugacy classes & character table

1342 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 507);