#27299860

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#352 ⋊ C₂
Sylow	C₅₃ ⋊ #12754

Subgroups

	subgroup	G/H
center	C₂	#20274661
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#352	C₂
socle	C₂×C₅₃	#1790
prime radical	#20288886	C₂
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₂×#352	D₅₃
hypocenter	C₅₃	#12754
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

782 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:

(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,15,18,30)(7,19,20,31)(9,23,17,21)(11,24,25,26)(13,27,29,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,27)(16,28)(17,25)(18,29)(21,26)(22,31)(23,24)(30,32)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,29)(12,26)(14,25)(15,23,30,21)(16,22)(19,27,31,32)(24,28)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,e^b=egh,f^b=fg,i^b=i^52,d^c=dg,e^d=eh,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, 85336);