C₆₁×#7423 #29510856

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7423

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1347
commutator	C₂×C₁₆	C₂×C₄×C₆₁
Frattini, cosocle	C₂×M₃₂	C₂²×C₆₁
prime radical	C₂³×C₆₁	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂×M₃₂×C₆₁ ◁ G

140 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 125
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 2 generators on 125 points:

(1,2)(3,14,16,45)(4,20,23,44)(5,27)(6,30)(7,33)(8,13)(9,15)(10,24,39,25)(11,22)(12,29)(17,50,18,51)(19,52,34,46)(21,41)(26,59,57,53)(28,62,31,38)(32,42)(35,60)(36,54)(37,49)(40,58,63,56)(43,48)(47,64)(55,61)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,8,16)(2,9,13,37)(4,21,23,47)(5,28,6,31)(7,34,29,19)(10,38,39,62)(11,41,42,64)(12,43,33,36)(14,44,45,20)(15,24,49,25)(17,22,18,32)(26,60,57,55)(27,61,30,35)(40,51,63,50)(46,53,52,59)(48,58,54,56)

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 411);