#27299167

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#347 ⋊ C₂
Sylow	C₅₃ ⋊ #12609

Subgroups

	subgroup	G/H
center	C₄	C₄×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#347	C₂
socle	C₂×C₅₃	#1749
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#347	D₅₃
hypocenter	C₅₃	#12609
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×#347

444 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)(56,65,66,68)(57,69)(60,72,64,74)(62,75)(67,82,81,80)(70,79)(73,85,78,84)(77,83);(54,56,57,67,58,68,70,82,59,66,69,81,71,65,79,80)(55,60,62,73,63,74,77,85,61,64,75,78,76,72,83,84);(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)(56,64,66,60)(57,62)(58,63)(59,61)(65,74,68,72)(67,78,81,73)(69,75)(70,77)(71,76)(79,83)(80,85,82,84)

Polycyclic presentation:

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

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, 84643);