#27083696

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3068 ⋊ C₂
Sylow	C₄₇ ⋊ #60469

Subgroups

	subgroup	G/H
center	C₂	C₂×#121×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3068	C₂
socle	C₂×C₄₇	C₂²×#121
hypercenter	#3068	D₄₇
hypocenter	C₄₇	#60469
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3068

2628 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)(48,52,54,53)(49,57,59,58)(50,63,62,64)(51,66,67,68)(55,71,70,60)(56,72,73,65)(61,77,76,78)(69,79,75,74);(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)(50,62)(55,70)(60,71)(61,76)(63,64)(69,75)(74,79)(77,78);(48,51)(49,56)(50,61)(52,66)(53,68)(54,67)(55,69)(57,72)(58,65)(59,73)(60,74)(62,76)(63,77)(64,78)(70,75)(71,79);(48,50,52,63,54,62,53,64)(49,55,58,60,59,70,57,71)(51,61,66,77,67,76,68,78)(56,69,65,74,73,75,72,79);(48,49)(50,60)(51,65)(52,57)(53,58)(54,59)(55,63)(56,66)(61,75)(62,71)(64,70)(67,72)(68,73)(69,76)(74,77)(78,79)

Polycyclic presentation:

2^8,47|b^2=f,e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=dh,g^c=gh,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, 1036986);