#27083274

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3046 ⋊ C₂
Sylow	C₄₇ ⋊ #60427

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3046	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#15689920	C₂
hypercenter	#3046	D₄₇
hypocenter	C₄₇	#60427
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3046

2678 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)(53,61,63,62)(54,66,65,67)(55,69,70,71)(59,74,73,75)(60,68,76,77)(64,80,79,81)(72,78,82,83);(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)(54,65)(59,73)(64,79)(66,67)(72,82)(74,75)(78,83)(80,81);(52,55)(53,60)(54,64)(56,69)(57,71)(58,70)(59,72)(61,68)(62,77)(63,76)(65,79)(66,80)(67,81)(73,82)(74,78)(75,83);(48,49,50,51)(52,54)(53,59)(55,64)(56,66)(57,67)(58,65)(60,72)(61,74)(62,75)(63,73)(68,78)(69,80)(70,79)(71,81)(76,82)(77,83);(49,51)(52,53,58,63)(54,59,65,73)(55,68,70,77)(56,62,57,61)(60,71,76,69)(64,78,79,83)(66,75,67,74)(72,81,82,80)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036564);