#27300326

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1378 ⋊ C₂
Sylow	C₅₃ ⋊ #12851

Subgroups

	subgroup	G/H
center	C₄	#16810485
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1378	C₂
socle	C₂×C₅₃	#1817
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1378	D₅₃
hypocenter	C₅₃	#12851
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1378

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,i^b=i^52,e^c=eh

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85802);