C₆₁×#10306 #29513739

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₆₁×#10306

Subgroups

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

670 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 89
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 89 points:

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

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 3294);