#27271979

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7540

Subgroups

	subgroup	G/H
center	C₂	#20272893
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting, prime radical	C₅₃×#2262	C₂
socle	C₂×C₅₃	#1415
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#2262	D₅₃
hypocenter	C₅₃	#7540
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55)(56,67)(57,73)(58,77)(59,80)(60,62)(61,81)(63,72)(64,74)(65,78)(66,83)(68,87)(69,102)(70,106)(71,84)(75,112)(76,82)(79,113)(85,116)(86,95)(88,111)(89,114)(90,100)(91,99)(92,105)(93,97)(94,96)(98,103)(101,115)(104,117)(107,110)(108,109);(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)(55,61,62,83)(56,66)(57,72)(58,74)(59,78)(63,89,87,97)(64,91,88,96)(65,92,80,86)(67,73,84,82)(68,76)(69,101)(70,105)(71,81)(75,100)(77,85,113,115)(79,111)(90,102,109,117)(93,94)(95,103)(98,112,106,110)(99,114)(104,116)(107,108)

Polycyclic presentation:

2^8,53|a^2=d,c^2=efh,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^52,c^b=ceh,d^b=dh,e^b=ef,f^b=fg,g^b=gh

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57455);