C₈₃×C₆₁⋊C₃ #29503371

order	15189
isoclinism	isoclinic class of C₆₁⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₈₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₈₃×C₆₁⋊C₃
unambiguous semidirect product	C₆₁⋊(C₃×C₈₃)
unambiguous semidirect product	(C₆₁×C₈₃)⋊C₃

Splittings

Fitting, socle, Hall	C₆₁×C₈₃ ⋊ C₃
derived, Sylow	C₆₁ ⋊ C₃×C₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₈₃	C₆₁⋊C₃
commutator, cosocle, hypocenter	C₆₁	C₃×C₈₃
Frattini	C₁	G
Fitting, socle	C₆₁×C₈₃	C₃
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₈₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 5063

A minimal faithful permutation representation — 2 generators on 144 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,34)(18,26,31)(21,29,28)(23,32,25)(27,37,44)(30,39,41)(33,42,38)(35,45,36)(40,48,52)(43,50,49)(46,53,47)(51,56,57)(54,58,55)(59,61,60)(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144);(1,2,4,8,15,24,35,46,54,59,58,42,20,5,9,16,25,36,37,14,23,33,43,51,53,32,12,19,28,38,47,48,26,7,13,21,30,40,45,22,31,41,49,55,56,39,17,3,6,11,18,27,34,44,52,57,60,61,50,29,10)

Conjugacy classes & character table

1909 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₈₃×C₆₁⋊C₃, D₈₃×C₆₁⋊C₃, C₈₃×D₆₁⋊C₃, …
maximal normal subgroup of	C₂×C₈₃×C₆₁⋊C₃, D₈₃×C₆₁⋊C₃, C₈₃×D₆₁⋊C₃, …
maximal quotient of	C₂×C₈₃×C₆₁⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15189, 1);

C83×C61⋊C3 #29503371