#49089881

order	35868
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇²×C₆₁ ⋊ C₂²×C₃
Hall	D₇²×C₆₁ ⋊ C₃
Hall	#22582226 ⋊ C₂²
Hall	D₇² ⋊ C₆₁⋊C₃
Sylow	C₆₁ ⋊ #64423⋊C₂
Sylow	C₇² ⋊ C₂²×C₆₁⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₆₁	C₂²×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 854

A minimal faithful permutation representation — 2 generators on 75 points:

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

Polycyclic presentation:

2^2,3,7^2,61|d^a=d^6,e^b=e^6,d^c=d^2,e^c=e^2,f^c=f^13

Conjugacy classes & character table

519 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35868, 233);