#27540879

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2192 ⋊ C₂
Sylow	C₅₃ ⋊ #33857

Subgroups

	subgroup	G/H
center	C₂	#20286385
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2192	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2192	D₅₃
hypocenter	C₅₃	#33857
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2192

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 117 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)(47,48)(49,50)(51,52)(54,57)(55,63)(56,70)(58,81)(59,77)(60,85)(61,78)(62,89)(64,96)(65,74)(66,99)(67,94)(68,102)(69,104)(71,106)(72,108)(73,109)(75,80)(76,83)(79,93)(82,114)(84,115)(86,105)(87,107)(88,110)(90,116)(91,112)(92,95)(97,113)(98,117)(100,101)(103,111);(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)(48,49)(50,51)(52,53)(54,57)(55,63)(56,69)(58,76)(59,85)(60,77)(61,78)(62,88)(64,93)(65,99)(66,74)(67,94)(68,101)(70,104)(71,105)(72,107)(73,75)(79,96)(80,109)(81,83)(82,115)(84,114)(86,106)(87,108)(89,110)(90,111)(91,92)(95,112)(97,117)(98,113)(100,102)(103,116);(54,56)(55,62)(57,75)(58,80)(59,72)(60,86)(61,73)(63,92)(64,95)(65,90)(66,100)(67,91)(68,79)(69,78)(70,76)(71,83)(74,111)(77,107)(81,108)(82,87)(84,104)(85,106)(88,94)(89,93)(96,116)(97,103)(98,110)(99,102)(101,117)(105,115)(109,114)(112,113);(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)(50,51)(52,53)(54,55,61,67)(56,68,73,103)(57,74,78,99)(58,79,84,97)(59,66,60,65)(62,87,91,71)(63,85,94,77)(64,82,98,83)(69,89,75,112)(70,88,109,92)(72,110,86,95)(76,113,114,93)(80,100,104,90)(81,96,115,117)(101,107,116,106)(102,108,111,105)

Polycyclic presentation:

2^8,53|a^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,i^a=i^52,c^b=ch,d^b=de,e^b=eg,g^b=gh,f^c=fh,i^c=i^52,e^d=eg,g^d=gh,i^d=i^52

Conjugacy classes & character table

641 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326355);