#27082564

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #60346

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting, prime radical	C₄₇×#3531	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#3531	D₄₇
hypocenter	C₄₇	#60346
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #3531

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 95 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)(64,67,68,78,69,79,80,91)(65,72,73,87,71,84,85,94)(66,75,76,89,74,77,88,90)(70,82,83,93,81,86,92,95);(49,55)(53,56)(54,63)(61,62)(66,74)(70,81)(75,77)(76,88)(82,86)(83,92)(89,90)(93,95);(48,51,52,60)(49,54,55,63)(50,57,58,59)(53,61,56,62)(65,71)(66,74)(72,84)(73,85)(75,77)(76,88)(87,94)(89,90);(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61)(64,66)(65,70)(67,77)(68,76)(69,74)(71,81)(72,86)(73,83)(75,79)(78,90)(80,88)(82,84)(85,92)(87,95)(89,91)(93,94);(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)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65,68,73,69,71,80,85)(66,70,76,83,74,81,88,92)(67,72,78,87,79,84,91,94)(75,82,89,93,77,86,90,95)

Polycyclic presentation:

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

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1035854);