#27092185

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3270 ⋊ C₂
Sylow	C₄₇ ⋊ #61081

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3270	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3270	D₄₇
hypocenter	C₄₇	#61081
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3270

2658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6016

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

Polycyclic presentation:

2^8,47|b^2=h,d^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^e=i^46

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045475);