#27082786

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3009 ⋊ C₂
Sylow	C₄₇ ⋊ #60370

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3009	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#15689958	C₂
hypercenter	#3009	D₄₇
hypocenter	C₄₇	#60370
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3009

4746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 83 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)(52,56,58,57)(53,61,63,62)(54,66,67,71)(55,69,68,70)(59,64,73,77)(60,75,74,76)(65,80,79,81)(72,78,82,83);(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)(55,68)(60,74)(65,79)(69,70)(72,82)(75,76)(78,83)(80,81);(49,51)(52,55)(53,60)(54,65)(56,69)(57,70)(58,68)(59,72)(61,75)(62,76)(63,74)(64,78)(66,80)(67,79)(71,81)(73,82)(77,83);(52,54)(53,59)(55,65)(56,66)(57,71)(58,67)(60,72)(61,64)(62,77)(63,73)(68,79)(69,80)(70,81)(74,82)(75,78)(76,83);(48,49)(50,51)(52,53,58,63)(54,64,67,77)(55,60,68,74)(56,62,57,61)(59,71,73,66)(65,78,79,83)(69,76,70,75)(72,81,82,80)

Polycyclic presentation:

2^8,47|a^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,d^c=dh,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, 1036076);