C₁₁×D₆₁ #13050708

order	1342
isoclinism	isoclinic class of D₆₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×D₆₁
unambiguous semidirect product	C₆₁⋊(C₂×C₁₁)

Splittings

Fitting, socle, Hall	C₁₁×C₆₁ ⋊ C₂
derived, Sylow	C₆₁ ⋊ C₂×C₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	D₆₁
commutator, cosocle, hypocenter	C₆₁	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁×C₆₁	C₂
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₁₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 671

A minimal faithful permutation representation — 2 generators on 72 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₁×#6923, C₁₁⋊#6923, D₁₁×D₆₁, …
maximal normal subgroup of	C₁₁×#6923, C₁₁⋊#6923, D₁₁×D₆₁, …
maximal quotient of	C₁₁×Dic₆₁, C₂×C₁₁×D₆₁, C₃×C₁₁×D₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1342, 2);

C11×D61 #13050708