#27301055

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#350 ⋊ C₂
Sylow	C₅₃ ⋊ #12996

Subgroups

	subgroup	G/H
center	C₂²	#16810759
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#350	C₂
socle	C₂²×C₅₃	#447
prime radical	C₂×#16810893	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#350	D₅₃
hypocenter	C₅₃	#12996
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#350

512 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)(58,60)(59,65)(61,75)(62,72)(63,73)(64,78)(66,85)(67,83)(68,84)(69,86)(70,77)(71,76)(74,87)(79,82)(80,89)(81,88);(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)(54,56,57,55)(59,64)(61,74)(62,76)(65,78)(66,67)(68,80)(69,82)(70,81)(71,72)(75,87)(77,88)(79,86)(83,85)(84,89);(54,55,57,56)(58,59,61,66,76,86,88,80,63,68,77,79,62,67,74,64)(60,69,70,78,71,84,87,85,73,83,75,89,72,65,81,82)

Polycyclic presentation:

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

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, 86531);