#27092215

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3011 ⋊ C₂
Sylow	C₄₇ ⋊ #61083

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3011	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3011	D₄₇
hypocenter	C₄₇	#61083
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3011

2674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045505);