#27475961

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#494 ⋊ C₂
Sylow	C₅₃ ⋊ #28498

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811173	C₂²
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#28498
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(58,62,65,74,64,73,63,60)(59,68,70,66,71,83,69,81)(61,77,79,76,80,88,78,72)(67,85,87,82,86,89,75,84);(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)(55,57)(58,61)(59,67)(60,72)(62,77)(63,78)(64,80)(65,79)(66,82)(68,85)(69,75)(70,87)(71,86)(73,88)(74,76)(81,84)(83,89);(54,55,56,57)(58,60,63,73,64,74,65,62)(59,66,69,68,71,81,70,83)(61,76,78,77,80,72,79,88)(67,84,75,89,86,82,87,85);(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)(58,59,65,69,64,71,63,70)(60,68,62,81,74,83,73,66)(61,75,78,67,80,87,79,86)(72,84,88,85,76,82,77,89)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=egh,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,i^a=i^52,c^b=cg,f^c=fh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 261437);