#27295032

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1347 ⋊ C₂
Sylow	C₅₃ ⋊ #11856

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1347	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	#20285289	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1347	D₅₃
hypocenter	C₅₃	#11856
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1347

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

normal subgroups only

This group has 378 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)(54,57,58,66)(55,60,61,69)(56,63,64,65)(59,67,62,68)(70,72,80,85,74,81,73,79)(71,76,82,84,78,83,77,75);(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)(55,59)(57,65)(58,64)(60,68)(61,62)(63,66)(67,69)(71,75)(72,79)(73,80)(76,77)(78,84)(81,85)(82,83);(54,55)(56,62,64,59)(57,60)(58,61)(63,68,65,67)(66,69)(70,71)(72,75)(73,82)(74,78)(76,79)(77,80)(81,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80508);