#27541192

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3336 ⋊ C₂
Sylow	C₅₃ ⋊ #33882

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3336	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20286437	C₂
hypercenter	#3336	D₅₃
hypocenter	C₅₃	#33882
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3336

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 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)(56,61)(58,60)(62,65,68,67)(63,70,72,82)(64,75,74,83)(66,77,81,80)(69,86,85,87)(71,78,89,92)(73,91,90,88)(76,79,84,93);(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)(62,64)(63,69)(65,75)(66,79)(67,83)(68,74)(70,86)(71,88)(72,85)(73,78)(76,80)(77,84)(81,93)(82,87)(89,91)(90,92);(54,56)(55,58)(57,61)(59,60)(64,74)(69,85)(73,90)(75,83)(76,84)(79,93)(86,87)(88,91);(54,55,57,59)(56,60,61,58)(62,63,68,72)(64,73,74,90)(65,70,67,82)(66,78,81,92)(69,84,85,76)(71,77,89,80)(75,91,83,88)(79,86,93,87)

Polycyclic presentation:

2^8,53|a^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh,i^c=i^52

Conjugacy classes & character table

953 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326668);