#27092267

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3278 ⋊ C₂
Sylow	C₄₇ ⋊ #61086

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3278	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3278	D₄₇
hypocenter	C₄₇	#61086
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3278

2666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6016

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

Polycyclic presentation:

2^8,47|b^2=h,d^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,e^c=eh,i^c=i^46,f^d=fh,i^e=i^46

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045557);