#27354972

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1452 ⋊ C₂
Sylow	C₅₃ ⋊ #17993

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#17993
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

576 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)(55,60)(56,62)(58,66)(59,65)(61,64)(63,69)(67,68)(70,74,76,72)(71,79,80,82)(73,81,75,83)(77,78,85,84);(54,56,58,61,59,63,67,55)(57,62,66,64,65,69,68,60)(70,73,76,75)(71,78,80,84)(72,81,74,83)(77,82,85,79);(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)(60,64)(62,69)(66,68)(70,72,76,74)(71,77,80,85)(73,81,75,83)(78,82,84,79);(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)(54,55,58,56,59,61,67,63)(57,64,66,69,65,60,68,62)(70,71,76,80)(72,79,74,82)(73,78,75,84)(77,83,85,81)

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=ch,d^a=dgh,e^a=eg,i^a=i^52,d^b=dh,e^b=eg,i^b=i^52,d^c=dg

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 140448);