C₆₁×#13851 #29517284

order	15616
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₆₁×#13851

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₆₁	C₂³
commutator	C₂³	C₂³×C₄×C₆₁
Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
socle	C₂⁴×C₆₁	C₂⁴
prime radical	C₂³×#41×C₆₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₁ ◁ G

4614 conjugacy classes of subgroups; only the 734 normal subgroups are shown.

normal subgroups only

This group has 4614 conjugacy classes of subgroups — too many to draw. Only its 734 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) = 81
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 4 generators on 81 points:

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

Conjugacy classes & character table

5368 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6839);