#27350264

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1466 ⋊ C₂
Sylow	C₅₃ ⋊ #17409

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1466	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#13133188	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1466	D₅₃
hypocenter	C₅₃	#17409
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1466

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg,i^b=i^52,d^c=dg,i^c=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135740);