#27537478

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33459

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	C₂²×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	#33459
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

674 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)(58,60,63,70)(59,65,68,69)(61,71,75,85)(62,72,74,86)(64,77,79,82)(66,80,78,83)(67,81,76,84)(73,87,88,89);(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)(60,70)(65,69)(71,85)(72,86)(77,82)(80,83)(81,84)(87,89);(59,64)(61,73)(62,74)(65,77)(66,67)(68,79)(69,82)(71,87)(72,86)(75,88)(76,78)(80,81)(83,84)(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)(54,55,57,56)(58,59)(60,69)(61,64)(62,76)(63,68)(65,70)(66,73)(67,74)(71,82)(72,81)(75,79)(77,85)(78,88)(80,89)(83,87)(84,86)

Polycyclic presentation:

2^8,53|a^2=f,c^2=f,d^2=h,e^2=gh,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,i^a=i^52,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, 322954);