#27092544

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3289 ⋊ C₂
Sylow	C₄₇ ⋊ #61107

Subgroups

	subgroup	G/H
center	C₂	D₄²×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3289	C₂
socle	C₂×C₄₇	C₂×D₄²
hypercenter	#3289	D₄₇
hypocenter	C₄₇	#61107
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3289

2624 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)(49,57)(54,71)(55,73)(56,74)(58,68)(59,72)(70,77)(75,76);(48,51)(49,56)(50,61)(52,66)(53,67)(54,70)(55,68)(57,74)(58,73)(59,76)(60,63)(62,79)(64,78)(65,69)(71,77)(72,75);(48,51)(49,55)(50,60)(52,65)(53,67)(54,70)(56,68)(57,73)(58,74)(59,75)(61,63)(62,78)(64,79)(66,69)(71,77)(72,76);(48,50,53,64)(49,54,57,71)(51,60,67,79)(52,63,69,62)(55,70,73,77)(56,75,74,76)(58,72,68,59)(61,66,78,65);(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)(50,59)(51,56)(52,68)(53,57)(54,62)(55,66)(58,69)(60,77)(61,76)(63,71)(64,72)(65,73)(67,74)(70,79)(75,78)

Polycyclic presentation:

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

Conjugacy classes & character table

650 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045834);