#27536857

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2120 ⋊ C₂
Sylow	C₅₃ ⋊ #33351

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#20288369	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2120	D₅₃
hypocenter	C₅₃	#33351
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,c^2=h,d^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fg,d^c=dh,i^d=i^52,f^e=fh

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322333);