C₆₁×#11491 #29514924

order	15616
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#11491

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₆₁×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

356 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 101
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,29,18,15,21,28,37,13,30,12,27)(11,22,25,38,24,17,32,34,26,36,40,33,31,35,23,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(2,5)(3,7)(9,11)(10,17)(12,23)(13,31)(14,25)(15,26)(16,24)(18,33)(19,36)(20,34)(21,35)(22,30)(27,38)(28,40)(29,32)(37,39);(9,12,13,28,15,29,16,14)(10,18,30,20,21,27,19,37)(11,23,31,40,26,32,24,25)(17,33,22,34,35,38,36,39)

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, 4479);