C₂⁴×#41×C₆₁ #29566530

order	15616
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₆₁ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂⁴×#41×C₆₁

Subgroups

	subgroup	G/H
center	C₂⁴×C₄×C₆₁	C₂²
commutator, Frattini, cosocle	C₂	C₂⁷×C₆₁
socle	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

65470 conjugacy classes of subgroups; only the 59038 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 59038 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) = 77
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 7 generators on 77 points:

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

Conjugacy classes & character table

9760 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 56085);