#27400096

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2898 ⋊ C₂
Sylow	C₅₃ ⋊ #22465

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2898	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#16811438	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2898	D₅₃
hypocenter	C₅₃	#22465
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2898

1312 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)(59,67)(60,72)(61,76)(63,79)(64,78)(65,80)(66,81)(68,85)(69,84)(70,86)(71,87)(73,88)(74,77)(75,89)(82,83);(58,61,65,59)(60,71,75,66)(62,76,80,67)(63,69,64,68)(70,73,82,74)(72,87,89,81)(77,86,88,83)(78,85,79,84);(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)(58,60)(59,66)(61,71)(62,77)(63,74)(64,73)(65,75)(67,83)(68,82)(69,70)(72,78)(76,86)(79,89)(80,88)(81,84)(85,87);(54,55,56,57)(58,59,63,69,65,61,64,68)(60,70,74,71,75,82,73,66)(62,67,79,84,80,76,78,85)(72,86,77,87,89,83,88,81)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185572);