#27091615

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3267 ⋊ C₂
Sylow	C₄₇ ⋊ #61051

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3267	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3267	D₄₇
hypocenter	C₄₇	#61051
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3267

2734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

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

Polycyclic presentation:

2^8,47|c^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,f^b=fh,f^d=fh,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, 1044905);