#27369243

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3218 ⋊ C₂
Sylow	C₅₃ ⋊ #19383

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133189
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3218	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20286077	C₂
hypercenter	#3218	D₅₃
hypocenter	C₅₃	#19383
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3218

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=eh,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,i^a=i^52,d^b=dg,i^b=i^52,d^c=dg,f^c=fh

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 154719);