C₆₁×#11908 #29515341

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#11908

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

388 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 85
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4896);