#27364783

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18943

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3552	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3552	D₅₃
hypocenter	C₅₃	#18943
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3552

846 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)(56,63)(60,67)(64,69)(66,68)(70,71)(72,76)(73,84)(74,78)(75,79)(77,81)(80,85)(82,83);(54,56)(55,60)(57,64)(58,63)(59,66)(61,68)(62,67)(65,69)(70,73)(71,77)(72,80)(74,82)(75,81)(76,78)(79,84)(83,85);(55,59)(57,65)(60,66)(61,62)(64,69)(67,68)(70,72,75,82)(71,76,79,83)(73,80,81,74)(77,78,84,85);(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)(54,55,58,62)(56,60,63,67)(57,59,65,61)(64,66,69,68)(70,71,75,79)(72,77,82,84)(73,83,81,76)(74,85,80,78)

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,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,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150259);