C₂×C₆₁×#1410 #29517020

order	15616
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₆₁×#1410

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#353
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₄⋊Q₈	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄⋊Q₈×C₆₁ ◁ G

450 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 79
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 79 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6575);