#27299706

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×AΣL(1,𝔽₁₆) ⋊ C₂
Sylow	C₅₃ ⋊ #12725

Subgroups

	subgroup	G/H
center	C₂	#20274631
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×AΣL(1,𝔽₁₆)	C₂
socle	C₂×C₅₃	#1782
prime radical	#20288402	C₂
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₂×AΣL(1,𝔽₁₆)	D₅₃
hypocenter	C₅₃	#12725
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×AΣL(1,𝔽₁₆)

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

(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)(55,60,61,62)(57,63)(58,64)(59,67,65,68)(66,69);(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)(55,59)(60,67)(61,65)(62,68);(54,55)(56,62)(57,65,66,59)(58,61)(60,64)(63,67,69,68)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fg,e^c=eh,i^c=i^52,f^d=fh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85182);