#27086309

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3178 ⋊ C₂
Sylow	C₄₇ ⋊ #60696

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3178	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20253056	C₂
hypercenter	#3178	D₄₇
hypocenter	C₄₇	#60696
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3178

1850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6016

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

Polycyclic presentation:

2^8,47|a^2=h,c^2=gh,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,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, 1039599);