#49089858

order	35868
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇²×C₆₁ ⋊ C₃×C₄
Hall	C₇²×#6923 ⋊ C₃
Hall	#22582226 ⋊ C₄
Hall	#6923 ⋊ #3861
Sylow	C₆₁ ⋊ C₄×#3861
Sylow	C₇² ⋊ #6923⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₆₁	C₃×C₄
prime radical	#31822466	C₂
cosocle	C₇²×D₆₁	C₂×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 2989

A minimal faithful permutation representation — 3 generators on 75 points:

(2,3,4)(5,7,6)(9,10,11)(12,14,13)(16,21,26,22,18,31,19,32,24,17,25,39)(20,41,46,42,28,30,29,52,43,27,44,47)(23,45,68,34,36,61,37,53,54,35,55,40)(33,56,74,67,50,69,51,70,64,49,62,60)(38,72,66,63,58,65,59,71,48,57,73,75);(8,9,10,12,11,13,14)(15,16,20,30,48,58,72,70,55,59,37,22,34,18,26,40,62,65,54,41,51,32,42,63,67,74,28,45,46,39,60,69,31,43,53,44,64,49,57,27,21,33,52,68,75,50,61,24,25,35,17,23,38,36,56,71,73,66,47,29,19);(1,2,3,5,4,6,7)

Polycyclic presentation:

2,3,2,7^2,61|a^2=c,f^a=f^50,d^b=d^2,e^b=e^2,f^b=f^13,f^c=f^60

Conjugacy classes & character table

321 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35868, 210);