#27299681

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2120 ⋊ C₂
Sylow	C₅₃ ⋊ #12720

Subgroups

	subgroup	G/H
center	C₄	#16810595
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	#1782
prime radical	#20288103	C₂
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2120	D₅₃
hypocenter	C₅₃	#12720
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(54,57)(55,61,64,78)(56,66)(58,71)(59,72)(60,74,76,77)(62,63,70,79)(65,80,81,75)(67,69)(68,82)(73,85)(83,84);(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)(54,56)(55,60)(57,67)(58,68)(59,69)(61,77)(62,65)(63,75)(64,76)(66,72)(70,81)(71,83)(73,84)(74,78)(79,80)(82,85);(54,55)(56,65)(57,70,72,62)(58,63)(59,64)(60,67,76,66)(61,71,78,85)(68,77)(69,81)(73,79)(74,84)(75,83,80,82)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,f^b=fg,f^c=fh,i^c=i^52,f^d=fh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85157);