#27085536

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3137 ⋊ C₂
Sylow	C₄₇ ⋊ #60612

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3137	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20256720	C₂
hypercenter	#3137	D₄₇
hypocenter	C₄₇	#60612
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3137

2050 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)(57,65,67,66)(58,71,70,72)(59,74,76,75)(63,79,78,68)(64,73,81,80)(69,84,83,85)(77,86,87,82);(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)(58,70)(63,78)(68,79)(69,83)(71,72)(77,87)(82,86)(84,85);(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)(49,53)(52,55)(56,59,60,74,62,76,61,75)(57,64,66,80,67,81,65,73)(58,69,71,84,70,83,72,85)(63,77,68,82,78,87,79,86);(48,50,51,54)(49,52,53,55)(56,58,61,72,62,70,60,71)(57,63,65,79,67,78,66,68)(59,69,75,85,76,83,74,84)(64,77,73,86,81,87,80,82);(48,49)(50,52)(51,53)(54,55)(56,57)(58,68)(59,73)(60,65)(61,66)(62,67)(63,71)(64,75)(69,77)(70,79)(72,78)(74,81)(76,80)(82,85)(83,87)(84,86)

Polycyclic presentation:

2^8,47|b^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=dh,i^c=i^46,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, 1038826);