#27421618

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1858 ⋊ C₂
Sylow	C₅₃ ⋊ #24421

Subgroups

	subgroup	G/H
center	C₂	#20278087
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1858	C₂
socle	C₂×C₅₃	#2410
prime radical	#20289496	C₂
hypercenter	#1858	D₅₃
hypocenter	C₅₃	#24421
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1858

1310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,57)(55,63)(56,67)(58,71)(59,72)(60,70)(61,69)(62,77)(64,79)(65,80)(66,81)(68,82)(73,83)(74,78)(75,84)(76,85);(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)(55,62)(57,67)(58,73)(59,68)(60,66)(61,75)(63,77)(64,76)(65,74)(69,84)(70,81)(71,83)(72,82)(78,80)(79,85);(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)(55,61)(56,66)(58,60)(62,76)(63,69)(65,75)(67,81)(68,73)(70,71)(77,85)(80,84)(82,83);(54,55,60,74,59,61,58,64)(56,65,73,62,68,76,66,75)(57,69,71,78,72,63,70,79)(67,80,81,84,82,85,83,77)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207094);