#27097649

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#464 ⋊ C₂
Sylow	C₄₇ ⋊ #61423

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#464	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	C₂×#464	D₄₇
hypocenter	C₄₇	#61423
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#464

1868 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)(57,64)(58,68)(63,78)(65,74)(67,83)(69,85)(70,79)(84,86);(48,51,52,50)(49,54,53,55)(58,68)(62,66)(67,83)(69,85)(76,80)(77,81)(82,87)(84,86);(48,50,52,51)(49,54,53,55)(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);(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)(48,49,52,53)(50,54,51,55)(56,58,60,68)(57,62,64,66)(59,67,71,83)(61,69,73,85)(63,76,78,80)(65,77,74,81)(70,87,79,82)(72,84,75,86);(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)(49,53)(54,55)(56,57)(58,66)(59,70)(60,64)(61,74)(62,68)(63,72)(65,73)(67,82)(69,77)(71,79)(75,78)(76,86)(80,84)(81,85)(83,87)

Polycyclic presentation:

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

Conjugacy classes & character table

929 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1050939);