C₆₁×H(3,𝔽₄) #16839455

order	3904
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₆₁×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₆₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ G

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 976

A minimal faithful permutation representation — 4 generators on 77 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,66,69)(64,71,72,68)(65,73,75,77)(67,70,76,74);(62,64,66,72)(63,68,69,71)(65,74,75,70)(67,73,76,77);(62,65)(63,64)(66,75)(67,68)(69,72)(70,73)(71,76)(74,77);(62,64)(63,65)(66,72)(67,73)(68,70)(69,75)(71,74)(76,77)

Conjugacy classes & character table

1159 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#30598673, #30598679, #30598683, …
maximal subgroup of	C₆₁×#1655, C₆₁×#1659, C₆₁×#2199, …
maximal normal subgroup of	C₆₁×#1655, C₆₁×#1659, C₆₁×#2199, …
maximal quotient of	C₆₁×#2375, C₆₁×#2379, C₆₁×#2670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3904, 242);

C61×H(3,𝔽4) #16839455