#27080883

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2922 ⋊ C₂
Sylow	C₄₇ ⋊ #60212

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₄₇×#2922	C₂
prime radical	#20256773	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#2922	D₄₇
hypocenter	C₄₇	#60212
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #2922

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 87 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)(59,69)(63,75)(66,78)(70,81)(72,82)(76,85)(79,86)(83,87);(48,50,51,54)(49,53,52,55)(58,67)(62,73)(66,78)(68,80)(72,82)(74,84)(79,86)(83,87);(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)(56,59,60,70,61,69,71,81)(57,63,64,76,65,75,77,85)(58,66,68,79,67,78,80,86)(62,72,74,83,73,82,84,87);(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)(49,52)(53,55)(56,58)(57,62)(59,66)(60,68)(61,67)(63,72)(64,74)(65,73)(69,78)(70,79)(71,80)(75,82)(76,83)(77,84)(81,86)(85,87);(48,49)(50,53)(51,52)(54,55)(56,57,60,64,61,65,71,77)(58,62,68,74,67,73,80,84)(59,63,70,76,69,75,81,85)(66,72,79,83,78,82,86,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1034173);