#27357289

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1468 ⋊ C₂
Sylow	C₅₃ ⋊ #18227

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1468	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288928	C₂
hypercenter	#1468	D₅₃
hypocenter	C₅₃	#18227
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1468

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(65,76)(69,83)(73,87)(77,89)(81,92)(84,90)(86,93)(88,91);(54,56)(55,58)(57,60)(59,61)(62,65)(63,69)(64,73)(66,77)(67,76)(68,81)(70,84)(71,83)(72,86)(74,88)(75,87)(78,90)(79,91)(80,89)(82,92)(85,93);(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)(56,58)(57,59)(60,61)(62,64,67,75)(63,68,71,82)(65,73,76,87)(66,79,80,74)(69,81,83,92)(70,85,78,72)(77,91,89,88)(84,93,90,86);(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)(58,61)(62,63,67,71)(64,72,75,85)(65,69,76,83)(66,78,80,70)(68,79,82,74)(73,86,87,93)(77,90,89,84)(81,91,92,88)

Polycyclic presentation:

2^8,53|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,i^a=i^52,e^b=eg,f^b=fh,i^b=i^52,d^c=dg,e^c=eh

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 142765);