#27277389

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8693

Subgroups

	subgroup	G/H
center	C₂²	#16810481
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1529	C₂
socle	C₂²×C₅₃	#380
prime radical	#121×C₅₃	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1529	D₅₃
hypocenter	C₅₃	#8693
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1529

358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 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)(55,59)(56,61)(58,60)(62,65)(63,70)(64,74)(66,79)(67,82)(68,80)(69,83)(71,85)(72,78)(73,87)(75,90)(76,92)(77,91)(81,86)(84,93)(88,89);(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)(54,56,59,58)(55,60,57,61)(62,64,66,75,68,77,67,76)(63,69,71,73,72,84,81,88)(65,74,79,90,80,91,82,92)(70,83,85,87,78,93,86,89);(54,55,58,61,59,57,56,60)(62,63,67,71,68,72,66,81)(64,73,76,84,77,88,75,69)(65,78,79,85,80,70,82,86)(74,89,90,93,91,87,92,83)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62865);