C₆₁×#7415 #29510848

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7415

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1345
commutator	C₂×M₁₆	C₂×C₄×C₆₁
Frattini, cosocle	#423	C₂²×C₆₁
prime radical	C₆₁×#575	C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#423 ◁ G

190 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 93 points:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 403);