#27085307

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #60594

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
prime radical	C₂×2₋¹⁺⁴×C₄₇	C₂²
hypercenter	#3011	D₄₇
hypocenter	C₄₇	#60594
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3011

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6016

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

Polycyclic presentation:

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

Conjugacy classes & character table

797 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1038597);