#27299849

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1889 ⋊ C₂
Sylow	C₅₃ ⋊ #12753

Subgroups

	subgroup	G/H
center	C₂	#20274662
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₅₃×#1889	C₂
socle	C₂×C₅₃	#1790
prime radical	#20288870	C₂
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	#1889	D₅₃
hypocenter	C₅₃	#12753
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #1889

936 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:

(7,12)(9,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)

Polycyclic presentation:

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

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, 85325);