#27300975

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1378 ⋊ C₂
Sylow	C₅₃ ⋊ #12981

Subgroups

	subgroup	G/H
center	C₈	#13133187
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1378	C₂
socle	C₂×C₅₃	#1869
prime radical	#16811135	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1378	D₅₃
hypocenter	C₅₃	#12981
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

488 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86451);