#27099097

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3269 ⋊ C₂
Sylow	C₄₇ ⋊ #61492

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3269	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3269	D₄₇
hypocenter	C₄₇	#61492
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3269

1770 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:

(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)(58,69)(63,79)(67,72)(68,70)(77,80)(78,85)(83,87)(84,86);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(49,52)(53,55)(56,60,62,61)(57,65,66,75)(58,68,69,70)(59,73,74,76)(63,78,79,85)(64,81,82,71)(67,84,72,86)(77,87,80,83);(56,59)(57,64)(58,67)(60,73)(61,76)(62,74)(63,77)(65,81)(66,82)(68,84)(69,72)(70,86)(71,75)(78,87)(79,80)(83,85);(48,49)(50,53)(51,52)(54,55)(56,58)(57,63)(59,72)(60,68)(61,70)(62,69)(64,80)(65,78)(66,79)(67,74)(71,87)(73,86)(75,85)(76,84)(77,82)(81,83);(49,52)(53,55)(56,57)(58,63)(59,71)(60,65)(61,75)(62,66)(64,73)(67,83)(68,78)(69,79)(70,85)(72,87)(74,81)(76,82)(77,84)(80,86)

Polycyclic presentation:

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

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1052387);