#27466290

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1812 ⋊ C₂
Sylow	C₅₃ ⋊ #27694

Subgroups

	subgroup	G/H
center	C₄	#16811192
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1812	C₂
socle	C₂×C₅₃	#2487
prime radical	#16811169	C₂²
hypercenter	#1812	D₅₃
hypocenter	C₅₃	#27694
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1812

626 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)(54,58,59,77,61,78,60,69)(55,64,65,76,67,80,66,83)(56,70,71,79,72,57,74,75)(62,73,68,82,81,85,84,63);(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)(55,63)(56,69)(58,71)(59,75)(60,79)(61,70)(62,80)(64,81)(65,73)(66,85)(67,82)(68,83)(72,77)(74,78)(76,84);(54,56,59,71,61,72,60,74)(55,62,65,68,67,81,66,84)(57,69,75,58,70,77,79,78)(63,64,73,76,82,80,85,83);(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)(54,55,59,66,61,67,60,65)(56,68,71,62,72,84,74,81)(57,73,75,63,70,85,79,82)(58,76,77,64,78,83,69,80)

Polycyclic presentation:

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

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, 251766);