C₂²×C₆₁×#354 #29536786

order	15616
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×C₆₁×#354

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#99
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₁
prime radical	C₂³×D₄×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁴×C₆₁ ◁ C₂³×D₄×C₆₁ ◁ G

1502 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 382 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) = 81
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 4 generators on 81 points:

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

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, 26341);