#27293776

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1326 ⋊ C₂
Sylow	C₅₃ ⋊ #11578

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1326	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×C₂²⋊D₅₃	C₂×C₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#1326	D₅₃
hypocenter	C₅₃	#11578
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1326

426 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 71 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79252);