#27299646

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 ⋊ #12713

Subgroups

	subgroup	G/H
center	C₂	#20274628
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	#1781
prime radical	#20288357	C₂
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2120	D₅₃
hypocenter	C₅₃	#12713
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

986 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)(55,62)(58,73)(59,75)(61,76)(63,65)(64,66)(67,81)(69,83)(70,85)(72,80)(77,79)(78,82);(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)(54,56,60,71)(55,61,66,79)(57,68,74,84)(58,69,73,83)(59,70,75,85)(62,76,64,77)(63,67,72,82)(65,81,80,78);(54,55,59,65)(56,67,70,76)(57,72,73,62)(58,64,74,63)(60,66,75,80)(61,69,78,84)(68,79,83,81)(71,82,85,77)

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fg,e^c=eh,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, 85122);