#27274499

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8125

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133124
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#432	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#432	D₅₃
hypocenter	C₅₃	#8125
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#432

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(71,76)(75,82)(77,81)(83,85);(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)(54,56)(55,59,61,62)(57,64)(58,63)(60,67,68,69)(65,66)(70,72,74,79,73,78,80,84)(71,75,77,83,76,82,81,85);(54,55,57,60,58,61,65,68)(56,62,64,69,63,59,66,67)(70,71,73,76)(72,75)(74,81,80,77)(78,82)(79,85)(83,84)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,e^2=gh,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg,i^b=i^52

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59975);