#120×C₁₁₃ #16813503

order	3616
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#120×C₁₁₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₁₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ G

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 904

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

1582 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#28406583, #28406585, #28406623, …
maximal subgroup of	AΣL(1,𝔽₁₆)×C₁₁₃, B(4,𝔽₂)×C₁₁₃, C₂×#120×C₁₁₃, …
maximal normal subgroup of	AΣL(1,𝔽₁₆)×C₁₁₃, B(4,𝔽₂)×C₁₁₃, C₂×#120×C₁₁₃, …
maximal quotient of	#379×C₁₁₃, #385×C₁₁₃, #386×C₁₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3616, 27);

#120×C113 #16813503