#27271561

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1398 ⋊ C₂
Sylow	C₅₃ ⋊ #7389

Subgroups

	subgroup	G/H
center	C₂	#20272683
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	#346	C₂×D₅₃
Fitting	C₅₃×#1398	C₂
socle	C₂×C₅₃	#1333
prime radical	#16811055	C₄
cosocle	C₅₃×#346	C₂²
hypercenter	#1398	D₅₃
hypocenter	C₅₃	#7389
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1398

118 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(54,55)(56,66)(57,71)(58,61)(59,63)(60,73)(62,79)(64,72)(65,75)(67,80)(68,83)(69,76)(70,74)(77,85)(78,81)(82,84);(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)(55,60,61,75,63,77,64,78)(56,65,68,81)(57,70)(58,72)(62,66,71,82,74,83,80,76)(67,79)(69,85,84,73)

Polycyclic presentation:

2^8,53|a^2=d,c^2=eh,d^2=f,e^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,c^b=ceg,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^52,d^c=dh

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57037);