#27103074

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3551 ⋊ C₂
Sylow	C₄₇ ⋊ #61666

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3551	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3551	D₄₇
hypocenter	C₄₇	#61666
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #532 ◁ #3551

2064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

926 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1056364);