#27304450

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1413 ⋊ C₂
Sylow	C₅₃ ⋊ #13598

Subgroups

	subgroup	G/H
center	C₂	#354×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₅₃
Fitting	C₅₃×#1413	C₂
socle	C₂×C₅₃	C₂×#354
prime radical	#16811174	C₂²
cosocle	C₄⋊Q₈×C₅₃	C₂³
hypercenter	#1413	D₅₃
hypocenter	C₅₃	#13598
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #1413

318 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89926);