C₆₁×D₁₁₁ #27214366

order	13542
isoclinism	isoclinic class of D₁₁₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₁×D₁₁₁
unambiguous semidirect product	C₃⋊(C₆₁×D₃₇)
unambiguous semidirect product	C₃₇⋊(D₃×C₆₁)

Splittings

Fitting, socle, Hall	C₃×C₃₇×C₆₁ ⋊ C₂
derived, Hall	C₃×C₃₇ ⋊ C₂×C₆₁
Hall	C₃₇×C₆₁ ⋊ D₃
Hall	C₃×C₆₁ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₃×C₆₁
Sylow	C₃ ⋊ C₆₁×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₁	D₁₁₁
commutator, cosocle, hypocenter	C₃×C₃₇	C₂×C₆₁
Frattini	C₁	G
Fitting, socle	C₃×C₃₇×C₆₁	C₂
LCS	G ▷ C₃×C₃₇
UCS	C₁ ◁ C₆₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6771

A minimal faithful permutation representation — 2 generators on 101 points:

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

Conjugacy classes & character table

3477 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₁×D₁₁₁, D₃×C₆₁×D₃₇, D₁₁₁⋊D₆₁, …
maximal normal subgroup of	D₆₁×D₁₁₁, D₃×C₆₁×D₃₇, D₁₁₁⋊D₆₁, …
maximal quotient of	C₆₁×Dic₁₁₁, C₂×C₆₁×D₁₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13542, 11);

C61×D111 #27214366