C₆₁×#12221 #29515654

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12221

Subgroups

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

392 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 93 points:

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5209);