#27276565

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1986 ⋊ C₂
Sylow	C₅₃ ⋊ #8512

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1986	C₂
socle	C₂²×C₅₃	#379
prime radical	#13133191	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1986	D₅₃
hypocenter	C₅₃	#8512
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1986

652 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,i^b=i^52,d^c=dgh,e^c=eg,i^c=i^52

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62041);