C₆₁×C₇₃⋊C₃ #27211342

order	13359
isoclinism	isoclinic class of C₇₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₁×C₇₃⋊C₃
unambiguous semidirect product	C₇₃⋊(C₃×C₆₁)

Splittings

Fitting, socle, Hall	C₆₁×C₇₃ ⋊ C₃
derived, Sylow	C₇₃ ⋊ C₃×C₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₆₁	C₇₃⋊C₃
commutator, cosocle, hypocenter	C₇₃	C₃×C₆₁
Frattini	C₁	G
Fitting, socle	C₆₁×C₇₃	C₃
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₆₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 4453

A minimal faithful permutation representation — 2 generators on 134 points:

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

Conjugacy classes & character table

1647 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₆₁×C₇₃⋊C₃, C₆₁×D₇₃⋊C₃, D₆₁×C₇₃⋊C₃, …
maximal normal subgroup of	C₂×C₆₁×C₇₃⋊C₃, C₆₁×D₇₃⋊C₃, D₆₁×C₇₃⋊C₃, …
maximal quotient of	C₂×C₆₁×C₇₃⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13359, 2);

C61×C73⋊C3 #27211342