#27402575

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #22690

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#22690
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1514 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 188051);