C₆₁×#12470 #29515903

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#12470

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₂×D₄×C₆₁	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈×D₄×C₆₁ ◁ G

224 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 7808

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

(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,22)(12,20,23,24)(13,19,15,16)(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);(2,5)(6,7)(9,11,18,17,13,20,12,19,14,21,22,10,15,24,23,16);(2,6)(4,8)(5,7)

Conjugacy classes & character table

2989 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5458);