#27411892

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2045 ⋊ C₂
Sylow	C₅₃ ⋊ #23520

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2045	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×#13133236	C₂
hypercenter	#2045	D₅₃
hypocenter	C₅₃	#23520
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2045

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 points:

(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)(58,62,64,63)(59,67,68,78)(60,71,72,79)(61,75,77,76)(65,81,82,69)(66,84,73,88)(70,87,74,89)(80,86,83,85);(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)(55,57)(58,61,63,76,64,77,62,75)(59,66,67,84,68,73,78,88)(60,70,71,87,72,74,79,89)(65,80,69,85,82,83,81,86);(58,60)(59,65)(61,74)(62,71)(63,79)(64,72)(66,83)(67,81)(68,82)(69,78)(70,77)(73,80)(75,89)(76,87)(84,85)(86,88);(54,55)(56,57)(58,59)(60,69)(61,73)(62,67)(63,78)(64,68)(65,71)(66,77)(70,86)(72,81)(74,85)(75,88)(76,84)(79,82)(80,89)(83,87)

Polycyclic presentation:

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

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197368);