#27103036

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3301 ⋊ C₂
Sylow	C₄₇ ⋊ #61663

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3301	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3301	D₄₇
hypocenter	C₄₇	#61663
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3301

2064 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)(49,53)(52,54)(56,59,61,71)(57,64,66,80)(58,67,69,70)(60,72,73,85)(62,75,77,79)(63,78,65,81)(68,82,83,84)(74,86,76,87);(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)(50,55)(52,54)(59,71)(64,80)(67,70)(72,85)(75,79)(78,81)(82,84)(86,87);(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)(57,63)(60,73)(62,74)(64,78)(65,66)(68,83)(72,85)(75,86)(76,77)(79,87)(80,81)(82,84);(48,50)(49,52)(51,55)(53,54)(56,58,61,69)(57,62,66,77)(59,70,71,67)(60,68,73,83)(63,74,65,76)(64,79,80,75)(72,84,85,82)(78,87,81,86);(48,49)(50,54)(51,53)(52,55)(56,57)(58,62)(59,64)(60,63)(61,66)(65,73)(67,75)(68,74)(69,77)(70,79)(71,80)(72,78)(76,83)(81,85)(82,86)(84,87)

Polycyclic presentation:

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

Conjugacy classes & character table

857 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1056326);