C₆₁×#12286 #29515719

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12286

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#408 ◁ G

306 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 109
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 109 points:

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

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, 5274);