#27286856

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10303

Subgroups

	subgroup	G/H
center	C₂²	#16810531
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3255	C₂
socle	C₂²×C₅₃	#392
prime radical	C₂³×C₅₃	#127
hypercenter	#3255	D₅₃
hypocenter	C₅₃	#10303
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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)(54,57,60,56)(55,62,65,61)(58,69,59,67)(63,68,64,66)(70,71)(72,80)(73,77)(74,78)(75,84)(76,81)(79,82)(83,85);(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)(54,56,60,57)(55,61,65,62)(58,69,59,67)(63,68,64,66)(70,72,74,82)(71,76,78,85)(73,83,75,81)(77,80,84,79);(54,55,60,65)(56,66,57,68)(58,64,59,63)(61,69,62,67)(70,71,74,78)(72,79,82,80)(73,84,75,77)(76,83,85,81)

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72332);