#27302276

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#350 ⋊ C₂
Sylow	C₅₃ ⋊ #13196

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#350	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#13133237	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#350	D₅₃
hypocenter	C₅₃	#13196
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

430 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)(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)(58,61,63,72,62,80,65,64)(59,67,69,78,68,70,81,86)(60,73,75,76,74,79,77,82)(66,83,84,89,71,85,87,88);(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)(55,57)(58,60,72,82,65,77,61,79,62,74,64,76,63,75,80,73)(59,66,67,85,69,84,78,88,68,71,70,83,81,87,86,89);(54,55,56,57)(58,59,64,70,65,69,80,86,62,68,72,67,63,81,61,78)(60,71,76,85,77,87,73,88,74,66,82,83,75,84,79,89)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ch,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 87752);