#27356151

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1451 ⋊ C₂
Sylow	C₅₃ ⋊ #18093

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18093
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dg,e^a=eg,i^a=i^52,d^b=dh,e^b=eg,i^b=i^52,d^c=dgh

Conjugacy classes & character table

1514 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 141627);