#27496233

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1953 ⋊ C₂
Sylow	C₅₃ ⋊ #30236

Subgroups

	subgroup	G/H
center	C₂	#20280678
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1953	C₂
socle	C₂×C₅₃	#2644
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	#1953	D₅₃
hypocenter	C₅₃	#30236
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1953

446 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)(54,57,58,70,59,71,72,83)(55,62,63,77,61,75,64,76)(56,67,66,80,69,81,68,82)(60,73,65,79,74,84,78,85);(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)(57,70)(58,72)(60,65)(62,76)(67,81)(68,69)(71,83)(73,84)(74,78)(75,77);(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)(55,60,61,74)(57,67)(58,66)(59,69)(62,73,75,84)(63,65,64,78)(68,72)(70,80)(71,81)(76,85,77,79)(82,83);(54,55,58,63,59,61,72,64)(56,65,68,60,69,78,66,74)(57,62,70,77,71,75,83,76)(67,79,82,73,81,85,80,84)

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 281709);