C₆₁×#12775 #29516208

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#12775

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₆₁
prime radical	C₂²×D₄×C₆₁	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×Q₈×C₆₁ ◁ G

740 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 97
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 97 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5763);