#27512539

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #31419

Subgroups

	subgroup	G/H
center	C₂	#20281150
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2009	C₂
socle	C₂×C₅₃	#2688
prime radical	C₅₃×#575	C₂²
hypercenter	#2009	D₅₃
hypocenter	C₅₃	#31419
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2009

652 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)(54,58,60,78)(55,63,65,91)(56,68,69,76)(57,72,74,107)(59,82,83,77)(61,85,66,90)(62,88,89,113)(64,94,79,75)(67,96,97,105)(70,102,80,100)(71,104,93,112)(73,108,81,106)(84,110,87,111)(86,109,92,99)(95,114,103,117)(98,116,101,115);(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)(55,62)(56,67,69,97)(58,72)(59,81)(60,74)(61,84,66,87)(63,88)(64,93)(65,89)(68,96,76,105)(70,101,80,98)(71,79)(73,83)(75,104)(77,108)(78,107)(82,106)(85,110,90,111)(86,95,92,103)(91,113)(94,112)(99,117,109,114)(100,116,102,115);(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)(54,56,59,80,60,69,83,70)(55,61,64,92,65,66,79,86)(57,67,73,101,74,97,81,98)(58,76,77,100,78,68,82,102)(62,87,71,103,89,84,93,95)(63,90,75,99,91,85,94,109)(72,105,106,115,107,96,108,116)(88,110,112,117,113,111,104,114);(54,55)(56,66)(57,71)(58,75)(59,79)(60,65)(61,69)(62,73)(63,77)(64,83)(67,95)(68,99)(70,86)(72,88)(74,93)(76,109)(78,94)(80,92)(81,89)(82,91)(84,101)(85,100)(87,98)(90,102)(96,110)(97,103)(104,106)(105,111)(107,113)(108,112)(114,116)(115,117)

Polycyclic presentation:

2^8,53|b^2=ef,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,i^b=i^52,f^c=fh,g^c=gh,i^c=i^52,f^d=fh,i^d=i^52

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 298015);