C₆₁×#7468 #29510901

order	15616
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7468

Subgroups

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₆₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₆₁
socle	C₂²×C₆₁	C₂²⋊C₁₆
prime radical	C₂³×C₆₁	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

182 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 7808

A minimal faithful permutation representation — 2 generators on 97 points:

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

Conjugacy classes & character table

5368 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 456);