C₆₁×#13302 #29516735

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#13302

Subgroups

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

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

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, 6290);