#27086128

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3175 ⋊ C₂
Sylow	C₄₇ ⋊ #60682

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3175	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3175	D₄₇
hypocenter	C₄₇	#60682
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3175

1878 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)(57,65,66,75)(58,69,68,70)(59,72,74,73)(63,78,77,84)(64,71,79,85)(67,82,81,83)(76,80,86,87);(48,50,51,54)(49,53,52,55)(58,68)(63,77)(67,81)(69,70)(76,86)(78,84)(80,87)(82,83);(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)(56,59,61,73,62,74,60,72)(57,64,65,71,66,79,75,85)(58,67,70,83,68,81,69,82)(63,76,78,80,77,86,84,87);(49,52)(53,55)(56,58)(57,63)(59,67)(60,69)(61,70)(62,68)(64,76)(65,78)(66,77)(71,80)(72,82)(73,83)(74,81)(75,84)(79,86)(85,87);(48,49)(50,53)(51,52)(54,55)(56,57)(58,63)(59,71)(60,65)(61,75)(62,66)(64,73)(67,80)(68,77)(69,78)(70,84)(72,79)(74,85)(76,83)(81,87)(82,86)

Polycyclic presentation:

2^8,47|c^2=g,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=dh,i^c=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, 1039418);