#27083223

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3045 ⋊ C₂
Sylow	C₄₇ ⋊ #60422

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3045	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3045	D₄₇
hypocenter	C₄₇	#60422
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3045

2714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 83 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(52,56,58,57)(53,61,62,70)(54,65,64,66)(55,68,69,71)(59,74,73,80)(60,75,76,67)(63,79,78,81)(72,83,82,77);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(54,64)(59,73)(63,78)(65,66)(72,82)(74,80)(77,83)(79,81);(52,55)(53,60)(54,63)(56,68)(57,71)(58,69)(59,72)(61,75)(62,76)(64,78)(65,79)(66,81)(67,70)(73,82)(74,83)(77,80);(48,49,50,51)(52,54)(53,59)(55,63)(56,65)(57,66)(58,64)(60,72)(61,74)(62,73)(67,77)(68,79)(69,78)(70,80)(71,81)(75,83)(76,82);(49,51)(52,53)(54,59)(55,67)(56,61)(57,70)(58,62)(60,68)(63,77)(64,73)(65,74)(66,80)(69,75)(71,76)(72,79)(78,83)(81,82)

Polycyclic presentation:

2^8,47|b^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=dh,g^c=gh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036513);