#27537439

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#358 ⋊ C₂
Sylow	C₅₃ ⋊ #33455

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#358	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133239	C₂²
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#358	D₅₃
hypocenter	C₅₃	#33455
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

660 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)(60,68)(65,79)(69,82)(70,83)(77,88)(80,86)(84,87)(85,89);(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)(54,56,57,55)(58,60)(59,65)(61,69)(62,70)(63,68)(64,77)(66,80)(67,79)(71,84)(72,85)(73,83)(74,82)(75,86)(76,87)(78,88)(81,89);(58,59,63,67)(60,65,68,79)(61,72,74,81)(62,75,73,66)(64,76,78,71)(69,85,82,89)(70,86,83,80)(77,87,88,84);(54,55,57,56)(59,64)(61,71)(62,73)(65,77)(66,81)(67,78)(69,84)(70,83)(72,75)(74,76)(79,88)(80,89)(82,87)(85,86)

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322915);