#49089859

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	#22582225 ⋊ 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	#31822467	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)(20,30,56,40,28,44,29,45,41,27,42,58)(23,49,67,33,35,57,36,38,50,34,51,43)(37,60,74,55,46,73,53,69,61,52,62,75)(47,66,54,71,64,68,59,48,70,63,72,65);(8,9,10,12,11,13,14)(15,16,20,30,46,61,73,65,67,53,36,22,33,24,25,38,54,60,35,44,59,39,40,55,71,72,41,57,42,32,48,66,21,28,43,56,64,63,52,27,31,47,58,51,69,70,49,18,26,34,17,23,37,50,68,75,74,62,45,29,19);(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^47,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, 211);