#27122807

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3417 ⋊ C₂
Sylow	C₄₇ ⋊ #62635

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₈×D₄₇
commutator, cosocle	C₈×C₄₇	C₂⁵
Frattini	C₈	C₂⁴×D₄₇
Fitting	C₄₇×#3417	C₂
socle	C₂×C₄₇	C₂³×D₈
hypercenter	#3417	D₄₇
hypocenter	C₄₇	#62635
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #3417

2038 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)(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)(48,52,55,54)(49,58,60,75)(50,63,65,76)(51,67,66,69)(53,70,74,73)(56,78,80,79)(57,82,81,98)(59,71,85,100)(61,87,88,84)(62,90,89,99)(64,72,92,101)(68,94,93,97)(77,103,102,104)(83,95,105,110)(86,108,107,106)(91,96,109,111);(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)(51,66)(57,81)(62,89)(67,69)(68,93)(77,102)(82,98)(83,105)(86,107)(90,99)(91,109)(94,97)(95,110)(96,111)(103,104)(106,108);(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)(49,57)(50,62)(52,67)(53,68)(54,69)(55,66)(56,77)(58,82)(59,83)(60,81)(61,86)(63,90)(64,91)(65,89)(70,94)(71,95)(72,96)(73,97)(74,93)(75,98)(76,99)(78,103)(79,104)(80,102)(84,106)(85,105)(87,108)(88,107)(92,109)(100,110)(101,111);(48,50)(49,56)(51,62)(52,63)(53,72)(54,76)(55,65)(57,77)(58,78)(59,84)(60,80)(61,71)(64,73)(66,89)(67,90)(68,96)(69,99)(70,92)(74,101)(75,79)(81,102)(82,103)(83,106)(85,87)(86,95)(88,100)(91,97)(93,111)(94,109)(98,104)(105,108)(107,110);(48,49)(50,61)(51,57)(52,58)(53,71)(54,75)(55,60)(56,64)(59,73)(62,86)(63,87)(65,88)(66,81)(67,82)(68,95)(69,98)(70,85)(72,78)(74,100)(76,84)(77,91)(79,101)(80,92)(83,97)(89,107)(90,108)(93,110)(94,105)(96,103)(99,106)(102,109)(104,111)

Polycyclic presentation:

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

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076097);