C₆₁×#12086 #29515519

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12086

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂×D₁₆×C₆₁	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×C₈×C₆₁ ◁ C₈²×C₆₁ ◁ G

234 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

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, 5074);