#27295301

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#364 ⋊ C₂
Sylow	C₅₃ ⋊ #11920

Subgroups

	subgroup	G/H
center	C₄	#16810366
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	C₂×#331
prime radical	#20285143	C₂
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#11920
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

400 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)(54,57,59,71)(55,61,64,63)(56,66,69,68)(58,70,73,81)(60,74,76,65)(62,77,80,79)(67,72,84,83)(75,78,85,82);(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)(55,60)(57,66)(58,72)(59,69)(61,74)(62,78)(63,65)(64,76)(67,81)(68,71)(70,84)(73,83)(75,79)(77,85)(80,82);(54,55,58,62,57,61,70,77,59,64,73,80,71,63,81,79)(56,65,67,82,68,76,83,85,69,74,84,78,66,60,72,75)

Polycyclic presentation:

2^8,53|a^2=e,c^2=h,d^2=g,e^2=fg,f^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,i^b=i^52

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80777);