C₆₁×#11953 #29515386

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#11953

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

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

normal subgroups only

This group has 296 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) = 109
transitive degree	μ_t(G) = 7808

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

(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,27,35,43,36,45,23,25,20,26,22,28,38,47)(19,29,31,42,39,46,34,24,33,41,30,48,32,44,37,40)(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)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,33)(18,24,25,40)(20,34,21,37)(22,39,35,32)(26,44,27,46)(28,48,43,42)(29,47,41,45)(30,38,31,36);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,25)(24,40)(26,27)(28,43)(29,41)(42,48)(44,46)(45,47)

Conjugacy classes & character table

2806 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4941);