D₃₅×C₆₁⋊C₃ #27165482

order	12810
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃₅×C₆₁⋊C₃
unambiguous semidirect product	C₅⋊(D₇×C₆₁⋊C₃)

Splittings

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₆₁	C₂×C₃
LCS	G ▷ C₅×C₇×C₆₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 2135

A minimal faithful permutation representation — 2 generators on 73 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)(63,64)(65,66)(67,68)(70,71)(72,73);(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)(62,63,65,67,68,66,64)(69,70,72,73,71)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃₅×D₆₁⋊C₃, (C₅⋊(D₆₁⋊C₃))⋊D₇, D₅×D₇×C₆₁⋊C₃, …
maximal normal subgroup of	D₃₅×D₆₁⋊C₃, (C₅⋊(D₆₁⋊C₃))⋊D₇, D₅×D₇×C₆₁⋊C₃, …
maximal quotient of	Dic₃₅×C₆₁⋊C₃, C₂×D₃₅×C₆₁⋊C₃, D₃₅×C₆₁⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 53);

Fitting, socle, derived, Hall	C₅×C₇×C₆₁ ⋊ C₂×C₃
Hall	C₅×C₇×C₆₁⋊C₃ ⋊ C₂
Hall	C₆₁×D₃₅ ⋊ C₃
Hall	C₇×C₆₁⋊C₃ ⋊ D₅
Hall	C₅×C₆₁⋊C₃ ⋊ D₇
Hall	C₇×C₆₁ ⋊ C₃×D₅
Hall	C₅×C₆₁ ⋊ C₃×D₇
Sylow	C₆₁ ⋊ C₃×D₃₅
Hall	C₅×C₇ ⋊ C₂×C₆₁⋊C₃
Sylow	C₇ ⋊ D₅×C₆₁⋊C₃
Sylow	C₅ ⋊ D₇×C₆₁⋊C₃

D35×C61⋊C3 #27165482