#27083529

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3062 ⋊ C₂
Sylow	C₄₇ ⋊ #60453

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3062	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3062	D₄₇
hypocenter	C₄₇	#60453
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3062

2722 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)(53,61,62,71)(54,66,65,67)(55,69,68,70)(59,63,73,77)(60,75,74,81)(64,78,80,79)(72,76,82,83);(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)(54,65)(55,68)(59,73)(60,74)(63,77)(66,67)(69,70)(75,81);(49,51)(52,55)(53,60)(54,64)(56,69)(57,70)(58,68)(59,72)(61,75)(62,74)(63,76)(65,80)(66,78)(67,79)(71,81)(73,82)(77,83);(52,54,57,67,58,65,56,66)(53,59,61,63,62,73,71,77)(55,64,70,79,68,80,69,78)(60,72,75,76,74,82,81,83);(48,49)(50,51)(52,53)(54,63)(55,60)(56,61)(57,71)(58,62)(59,67)(64,76)(65,77)(66,73)(68,74)(69,75)(70,81)(72,79)(78,82)(80,83)

Polycyclic presentation:

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

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036819);