#27411070

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2310 ⋊ C₂
Sylow	C₅₃ ⋊ #23408

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2310	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288048	C₂
hypercenter	#2310	D₅₃
hypocenter	C₅₃	#23408
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2310

1946 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)(60,72)(65,80)(70,82)(71,85)(73,77)(79,89)(84,88)(86,87);(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)(55,57)(58,61)(59,66)(60,71)(62,75)(63,78)(64,76)(65,79)(67,74)(68,83)(69,81)(70,84)(72,85)(73,86)(77,87)(80,89)(82,88);(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)(58,60)(59,65)(61,71)(62,77)(63,73)(64,72)(66,79)(67,82)(68,70)(69,80)(74,88)(75,87)(76,85)(78,86)(81,89)(83,84);(58,59,64,69)(60,70,72,82)(61,74,76,83)(62,68,63,67)(65,77,80,73)(66,78,81,75)(71,79,85,89)(84,86,88,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196546);