#27276992

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8610

Subgroups

	subgroup	G/H
center	C₂²	#16810481
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1799	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1799	D₅₃
hypocenter	C₅₃	#8610
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1799

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 69 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)(55,58)(57,61)(59,60)(63,67)(66,69);(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)(62,63,64,66,65,67,68,69);(54,55)(56,60,61,58)(57,59)(63,66)(64,68)(67,69)

Polycyclic presentation:

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

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, 62468);