#27083042

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3026 ⋊ C₂
Sylow	C₄₇ ⋊ #60401

Subgroups

	subgroup	G/H
center	C₂	C₂×#120×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₄₇×#3026	C₂
socle	C₂×C₄₇	C₂²×#120
hypercenter	#3026	D₄₇
hypocenter	C₄₇	#60401
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #3026

4854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1504

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

Polycyclic presentation:

2^8,47|a^2=h,b^a=bf,c^a=cg,i^a=i^46,g^b=gh,f^c=fh,e^d=eh,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, 1036332);