#27381395

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3387 ⋊ C₂
Sylow	C₅₃ ⋊ #20323

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3387	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#13133214	C₂³
hypercenter	#3387	D₅₃
hypocenter	C₅₃	#20323
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3387

572 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)(55,62)(60,67)(61,69)(64,68)(70,73,75,71)(72,78,79,85)(74,77,84,82)(76,83,81,80);(54,57,59,66)(55,61,62,69)(56,63,58,65)(60,68,67,64)(72,79)(76,81)(78,85)(80,83);(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)(54,56,59,58)(55,60,62,67)(57,65,66,63)(61,64,69,68)(70,72,75,79)(71,76,73,81)(74,83,84,80)(77,85,82,78);(54,55)(56,60)(57,64)(58,67)(59,62)(61,63)(65,69)(66,68)(70,71,75,73)(72,78,79,85)(74,82,84,77)(76,83,81,80)

Polycyclic presentation:

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

Conjugacy classes & character table

791 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 166871);