#27091165

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3237 ⋊ C₂
Sylow	C₄₇ ⋊ #61018

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3237	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#15689692	C₂²
hypercenter	#3237	D₄₇
hypocenter	C₄₇	#61018
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3237

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 3008

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

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1044455);