#27512568

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1902 ⋊ C₂
Sylow	C₅₃ ⋊ #31421

Subgroups

	subgroup	G/H
center	C₂	#20281149
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1902	C₂
socle	C₂×C₅₃	#2688
prime radical	#20275769	C₂
hypercenter	#1902	D₅₃
hypocenter	C₅₃	#31421
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1902

668 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)(54,58,60,62)(55,63,65,57)(56,68,69,74)(59,71,78,75)(61,67,66,80)(64,72,76,73)(70,85,77,82)(79,84,81,83);(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)(54,57,60,63)(55,62,65,58)(56,67)(59,73,78,72)(61,74)(64,75,76,71)(66,68)(69,80)(70,84)(77,83)(79,82)(81,85);(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)(54,56,59,77,60,69,78,70)(55,61,64,81,65,66,76,79)(57,67,72,84,63,80,73,83)(58,74,75,82,62,68,71,85);(54,55)(56,66)(57,71)(58,73)(59,76)(60,65)(61,69)(62,72)(63,75)(64,78)(67,82)(68,83)(70,79)(74,84)(77,81)(80,85)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 298044);