#27083048

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3027 ⋊ C₂
Sylow	C₄₇ ⋊ #60402

Subgroups

	subgroup	G/H
center	C₂	C₂×#120×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₄₇×#3027	C₂
socle	C₂×C₄₇	C₂²×#120
hypercenter	#3027	D₄₇
hypocenter	C₄₇	#60402
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3027

4778 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)(48,53,54,49)(50,60,61,55)(51,63,64,56)(52,57,65,66)(58,70,72,67)(59,68,73,74)(62,69,75,76)(71,77,79,78);(48,52,54,65)(49,57,53,66)(50,59,61,73)(51,62,64,75)(55,68,60,74)(56,69,63,76)(58,71,72,79)(67,77,70,78);(48,51)(49,56)(50,58)(52,62)(53,63)(54,64)(55,67)(57,69)(59,71)(60,70)(61,72)(65,75)(66,76)(68,77)(73,79)(74,78);(48,50)(49,55)(51,58)(52,59)(53,60)(54,61)(56,67)(57,68)(62,71)(63,70)(64,72)(65,73)(66,74)(69,77)(75,79)(76,78);(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)(48,49,54,53)(50,55,61,60)(51,56,64,63)(52,57,65,66)(58,70,72,67)(59,68,73,74)(62,69,75,76)(71,78,79,77)

Polycyclic presentation:

2^8,47|a^2=h,d^2=h,e^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, 1036338);