C₆₁×#7411 #29510844

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7411

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1344
commutator	C₂²×C₈	C₂×C₄×C₆₁
Frattini, cosocle	#408	C₂²×C₆₁
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#408 ◁ G

162 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 2 generators on 93 points:

(1,2,4,9,5,11,17,25)(3,8,13,22,14,23,30,24)(6,10,16,26,18,27,15,31)(7,19,20,12,21,28,32,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,7,11,21)(3,12)(4,15)(6,18)(8,10,23,27)(9,22,25,24)(14,29)(16,17)(19,28)(20,31,32,26)

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 399);