#27302607

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13262

Subgroups

	subgroup	G/H
center	C₂²	#16810818
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#13262
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 89 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88083);