#27085827

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3156 ⋊ C₂
Sylow	C₄₇ ⋊ #60648

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3156	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3156	D₄₇
hypocenter	C₄₇	#60648
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3156

3114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6016

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

Polycyclic presentation:

2^8,47|a^2=h,b^2=fh,c^2=g,f^2=h,b^a=bf,c^a=cg,f^a=fh,i^c=i^46,e^d=eh,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, 1039117);