#27465197

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1827 ⋊ C₂
Sylow	C₅₃ ⋊ #27598

Subgroups

	subgroup	G/H
center	C₄	#16811192
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1827	C₂
socle	C₂×C₅₃	#2482
prime radical	#16811459	C₂²
hypercenter	#1827	D₅₃
hypocenter	C₅₃	#27598
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1827

728 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)(54,56)(55,61)(57,66)(58,74)(59,67)(60,68)(62,75)(63,81)(64,76)(65,77)(69,84)(70,73)(71,82)(72,83)(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)(54,57)(55,62)(56,66)(58,73)(59,72)(60,71)(61,75)(63,80)(64,69)(65,79)(67,83)(68,82)(70,74)(76,84)(77,85)(78,81);(54,56)(55,61)(57,70,71,83)(58,74)(59,67)(60,68)(62,78,79,84)(63,81)(64,76)(65,77)(66,73,82,72)(69,75,80,85);(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)(54,55,59,64,60,65,58,63)(56,61,67,76,68,77,74,81)(57,69,73,62,71,80,72,79)(66,78,70,85,82,84,83,75)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 250673);