#27297367

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12290

Subgroups

	subgroup	G/H
center	C₂²	#16810758
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1347	C₂
socle	C₂²×C₅₃	#447
prime radical	C₅₃×#466	C₂²
hypercenter	#1347	D₅₃
hypocenter	C₅₃	#12290
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1347

316 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 61 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)(54,57,58,64)(55,60,61,67)(56,62,63,69)(59,65,66,68)(70,72,74,75)(71,77,78,83)(73,79,82,80)(76,81,84,85);(54,56)(55,59)(57,62)(58,63)(60,65)(61,66)(64,69)(67,68)(70,71)(72,77)(73,81)(74,78)(75,83)(76,80)(79,84)(82,85);(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)(54,55,58,61)(56,59,63,66)(57,60,64,67)(62,68,69,65)(71,76)(72,75)(73,80)(77,85)(78,84)(79,82)(81,83)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 82843);