#27281979

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9493

Subgroups

	subgroup	G/H
center	C₂²	#16810500
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1464	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1464	D₅₃
hypocenter	C₅₃	#9493
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1464

458 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 67455);