C₆₁×#12974 #29516407

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#12974

Subgroups

	subgroup	G/H
center	C₄×C₆₁	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×#41	C₂³×C₆₁
socle	C₂×C₆₁	#1867
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂×#41×C₆₁ ◁ G

408 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 114 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:

(1,2,7,12,8,13,6,23)(3,14,17,31,18,30,16,25)(4,11,19,10,5,20,22,29)(9,24,27,21,28,32,26,15)(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);(1,3,7,17,8,18,6,16)(2,9,12,27,13,28,23,26)(4,15,19,24,5,21,22,32)(10,25,20,14,29,31,11,30);(1,4)(2,10)(3,15)(5,8)(6,22)(7,19)(9,25)(11,23)(12,20)(13,29)(14,27)(16,32)(17,24)(18,21)(26,30)(28,31)

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, 5962);