#27091773

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3266 ⋊ C₂
Sylow	C₄₇ ⋊ #61060

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3266	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3266	D₄₇
hypocenter	C₄₇	#61060
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3266

2686 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)(53,60)(58,74)(59,76)(61,70)(62,73)(72,75)(77,82)(78,81);(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)(56,71)(58,73)(61,70)(62,74)(63,79)(65,66)(67,80)(68,83)(72,81)(75,78)(77,82);(49,51)(52,55,57,69)(53,59,60,76)(54,63,66,67)(56,68,71,83)(58,72,74,75)(61,77,70,82)(62,78,73,81)(64,79,65,80);(52,54,57,66)(53,58,60,74)(55,67,69,63)(56,64,71,65)(59,75,76,72)(61,73,70,62)(68,80,83,79)(77,78,82,81);(48,49)(50,51)(52,53)(54,62)(55,59)(56,70)(57,60)(58,65)(61,71)(63,78)(64,74)(66,73)(67,81)(68,82)(69,76)(72,80)(75,79)(77,83)

Polycyclic presentation:

2^8,47|b^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045063);