#27083715

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#472 ⋊ C₂
Sylow	C₄₇ ⋊ #60471

Subgroups

	subgroup	G/H
center	C₂	C₂×#121×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#472	C₂
socle	C₂×C₄₇	C₂²×#121
hypercenter	C₂×#472	D₄₇
hypocenter	C₄₇	#60471
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#472

2630 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037005);