#27288907

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10636

Subgroups

	subgroup	G/H
center	C₂²	#16810548
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1993	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂³×C₅₃	#124
hypercenter	#1993	D₅₃
hypocenter	C₅₃	#10636
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1993

278 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 87 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:

(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)(56,62)(58,65)(59,66)(63,64)(67,69)(70,72,73,82,75,81,74,83)(71,77,78,80,79,85,76,84);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,63)(55,59,60,67)(58,62,65,64)(61,69,68,66)(71,76)(72,81)(73,74)(77,84)(78,79)(80,85);(54,55)(56,59)(57,60)(58,61)(62,69)(63,67)(64,66)(65,68)(70,71,74,76,75,79,73,78)(72,80,82,85,81,84,83,77)

Polycyclic presentation:

2^8,53|a^2=eh,b^2=h,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=eg,f^c=fgh,i^c=i^52

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74383);