#27122821

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3414 ⋊ C₂
Sylow	C₄₇ ⋊ #62637

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₈×D₄₇
commutator, cosocle	C₈×C₄₇	C₂⁵
Frattini	C₈	C₂⁴×D₄₇
Fitting	C₄₇×#3414	C₂
socle	C₂×C₄₇	C₂³×D₈
hypercenter	#3414	D₄₇
hypocenter	C₄₇	#62637
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 111 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)(49,57)(55,76)(56,78)(58,81)(59,73)(60,86)(68,84)(75,101)(77,83)(79,103)(80,98)(82,92)(85,107)(95,106)(102,105)(104,110);(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)(51,65)(56,78)(61,87)(66,93)(67,94)(75,101)(79,103)(80,98)(85,107)(88,108)(89,99)(95,106)(96,111)(97,109)(102,105)(104,110);(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)(48,51)(49,56)(50,61)(52,66)(53,67)(54,65)(55,75)(57,78)(58,79)(59,80)(60,85)(62,88)(63,89)(64,87)(68,95)(69,96)(70,94)(71,97)(72,93)(73,98)(74,99)(76,101)(77,102)(81,103)(82,104)(83,105)(84,106)(86,107)(90,109)(91,108)(92,110)(100,111);(48,50,54,64)(49,55,57,76)(51,61,65,87)(52,69,72,100)(53,74,70,63)(56,75,78,101)(58,82,81,92)(59,83,73,77)(60,84,86,68)(62,90,91,71)(66,96,93,111)(67,99,94,89)(79,104,103,110)(80,105,98,102)(85,106,107,95)(88,109,108,97);(48,49)(50,60)(51,56)(52,68)(53,73)(54,57)(55,62)(58,71)(59,70)(61,85)(63,92)(64,86)(65,78)(66,95)(67,98)(69,83)(72,84)(74,82)(75,88)(76,91)(77,100)(79,97)(80,94)(81,90)(87,107)(89,110)(93,106)(96,105)(99,104)(101,108)(102,111)(103,109)

Polycyclic presentation:

2^8,47|b^2=h,f^2=gh,g^2=h,b^a=bf,e^a=eh,f^a=fg,g^a=gh,f^b=fg,g^b=gh,d^c=dh,i^c=i^46,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, 1076111);