C₆₁×#8797 #29512230

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#8797

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#382
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂³×C₆₁	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

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

normal subgroups only

This group has 228 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:

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

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1785);