#27082742

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3012 ⋊ C₂
Sylow	C₄₇ ⋊ #60366

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3012	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3012	D₄₇
hypocenter	C₄₇	#60366
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3012

4782 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)(52,56,58,57)(53,61,62,70)(54,65,66,71)(55,68,67,69)(59,73,74,63)(60,76,75,80)(64,79,78,81)(72,83,82,77);(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)(55,67)(60,75)(64,78)(68,69)(72,82)(76,80)(77,83)(79,81);(49,51)(52,55)(53,60)(54,64)(56,68)(57,69)(58,67)(59,72)(61,76)(62,75)(63,77)(65,79)(66,78)(70,80)(71,81)(73,83)(74,82);(52,54)(53,59)(55,64)(56,65)(57,71)(58,66)(60,72)(61,73)(62,74)(63,70)(67,78)(68,79)(69,81)(75,82)(76,83)(77,80);(48,49)(50,51)(52,53)(54,63)(55,60)(56,61)(57,70)(58,62)(59,65)(64,77)(66,73)(67,75)(68,76)(69,80)(71,74)(72,79)(78,83)(81,82)

Polycyclic presentation:

2^8,47|e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,d^c=dh,i^d=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, 1036032);