#27357902

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1500 ⋊ C₂
Sylow	C₅₃ ⋊ #18282

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1500	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20273662	C₂
hypercenter	#1500	D₅₃
hypocenter	C₅₃	#18282
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1500

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 143378);