#27092293

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3282 ⋊ C₂
Sylow	C₄₇ ⋊ #61088

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3282	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3282	D₄₇
hypocenter	C₄₇	#61088
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3282

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

Polycyclic presentation:

2^8,47|a^2=h,e^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,f^d=fh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045583);