#27096325

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

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #61331

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₄₇×#3045	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3045	D₄₇
hypocenter	C₄₇	#61331
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3045

1764 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)(56,60,61,62)(57,65,66,75)(58,70,69,74)(59,72,73,76)(63,78,67,79)(64,80,81,71)(68,85,84,86)(77,87,82,83);(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)(48,50,51,55)(49,53,52,54)(58,69)(63,67)(68,84)(70,74)(77,82)(78,79)(83,87)(85,86);(56,59)(57,64)(58,68)(60,72)(61,73)(62,76)(63,77)(65,80)(66,81)(67,82)(69,84)(70,85)(71,75)(74,86)(78,87)(79,83);(48,49,51,52)(50,54,55,53)(56,58,61,69)(57,63,66,67)(59,68,73,84)(60,74,62,70)(64,77,81,82)(65,79,75,78)(71,87,80,83)(72,86,76,85);(49,52)(53,54)(56,57)(58,67)(59,71)(60,65)(61,66)(62,75)(63,69)(64,72)(68,83)(70,79)(73,80)(74,78)(76,81)(77,85)(82,86)(84,87)

Polycyclic presentation:

2^8,47|b^2=f,d^2=fh,e^2=h,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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1049615);