C₆₁×#12591 #29516024

order	15616
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#12591

Subgroups

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

390 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 15616

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

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

Conjugacy classes & character table

6832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5579);