#27080197

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2905 ⋊ C₂
Sylow	C₄₇ ⋊ #60140

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₄₇×#2905	C₂
prime radical	#20256721	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#60140
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #2905

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:

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

Polycyclic presentation:

2^8,47|a^2=g,c^2=g,g^2=h,b^a=bf,e^a=eh,d^b=dh,i^b=i^46,i^c=i^46

Conjugacy classes & character table

1100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033487);