#27102450

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3200 ⋊ C₂
Sylow	C₄₇ ⋊ #61636

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3200	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3200	D₄₇
hypocenter	C₄₇	#61636
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3200

2074 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)(49,53)(52,54)(58,69)(63,78)(67,84)(68,73)(76,86)(77,79)(82,87)(83,85);(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)(50,55)(52,54)(56,60,62,61)(57,65,66,74)(58,68,69,73)(59,71,72,75)(63,77,78,79)(64,80,81,70)(67,83,84,85)(76,82,86,87);(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)(56,59)(57,64)(58,67)(60,71)(61,75)(62,72)(63,76)(65,80)(66,81)(68,83)(69,84)(70,74)(73,85)(77,82)(78,86)(79,87);(48,50)(49,52)(51,55)(53,54)(56,58,62,69)(57,63,66,78)(59,67,72,84)(60,73,61,68)(64,76,81,86)(65,79,74,77)(70,82,80,87)(71,85,75,83);(48,49)(50,54)(51,53)(52,55)(56,57)(58,63)(59,70)(60,65)(61,74)(62,66)(64,71)(67,82)(68,77)(69,78)(72,80)(73,79)(75,81)(76,85)(83,86)(84,87)

Polycyclic presentation:

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

Conjugacy classes & character table

650 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1055740);