#27085076

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3428 ⋊ C₂
Sylow	C₄₇ ⋊ #60581

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3428	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20252630	C₂
hypercenter	#3428	D₄₇
hypocenter	C₄₇	#60581
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #3428

2024 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)(59,70)(64,78)(67,80)(71,83)(74,85)(77,79)(81,87)(84,86);(48,50,51,54)(49,53,52,55)(56,59)(57,64)(58,67)(60,71)(61,70)(62,74)(63,77)(65,79)(66,78)(68,81)(69,80)(72,83)(73,84)(75,86)(76,85)(82,87);(57,63)(60,72)(62,73)(64,77)(65,66)(68,82)(71,83)(74,84)(75,76)(78,79)(81,87)(85,86);(49,52)(53,55)(56,58,61,69)(57,62,66,76)(59,67,70,80)(60,68,72,82)(63,73,65,75)(64,74,78,85)(71,81,83,87)(77,84,79,86);(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)(48,49)(50,53)(51,52)(54,55)(56,57,61,66)(58,62,69,76)(59,64,70,78)(60,63,72,65)(67,74,80,85)(68,73,82,75)(71,77,83,79)(81,84,87,86)

Polycyclic presentation:

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

Conjugacy classes & character table

1625 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1038366);