#27098919

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3328 ⋊ C₂
Sylow	C₄₇ ⋊ #61484

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3328	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#13093995	D₄
hypercenter	#3328	D₄₇
hypocenter	C₄₇	#61484
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3328

1750 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)(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)(48,50,51,54)(49,53,52,55)(58,68)(62,66)(67,83)(69,85)(76,80)(77,81)(82,87)(84,86);(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)(49,52)(53,55)(58,68)(62,66)(67,83)(69,85)(76,80)(77,81)(82,87)(84,86);(56,59,60,71)(57,63,64,78)(58,67,68,83)(61,75,73,72)(62,76,66,80)(65,79,74,70)(69,86,85,84)(77,82,81,87);(48,49)(50,53)(51,52)(54,55)(56,58)(57,62)(59,67)(60,68)(61,69)(63,76)(64,66)(65,77)(70,87)(71,83)(72,84)(73,85)(74,81)(75,86)(78,80)(79,82);(49,52)(53,55)(56,57,60,64)(58,66,68,62)(59,70,71,79)(61,74,73,65)(63,75,78,72)(67,82,83,87)(69,77,85,81)(76,84,80,86)

Polycyclic presentation:

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

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1052209);