#27093628

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3314 ⋊ C₂
Sylow	C₄₇ ⋊ #61173

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3314	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20257059	C₂
hypercenter	#3314	D₄₇
hypocenter	C₄₇	#61173
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3314

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 87 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)(56,60,62,61)(57,65,66,74)(58,68,69,73)(59,71,72,75)(63,77,79,78)(64,80,81,70)(67,83,85,84)(76,82,86,87);(50,55)(52,54);(56,59)(57,64)(58,67)(60,71)(61,75)(62,72)(63,76)(65,80)(66,81)(68,83)(69,85)(70,74)(73,84)(77,82)(78,87)(79,86);(48,50)(49,52)(51,55)(53,54)(56,58,62,69)(57,63,66,79)(59,67,72,85)(60,73,61,68)(64,76,81,86)(65,78,74,77)(70,82,80,87)(71,84,75,83);(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)(48,49,51,53)(50,54,55,52)(56,57)(58,63)(59,70)(60,65)(61,74)(62,66)(64,71)(67,82)(68,77)(69,79)(72,80)(73,78)(75,81)(76,84)(83,86)(85,87)

Polycyclic presentation:

2^8,47|a^2=f,b^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,i^a=i^46,d^b=df,e^b=eh,g^c=gh,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, 1046918);