#27294842

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1353 ⋊ C₂
Sylow	C₅₃ ⋊ #11818

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1353	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1353	D₅₃
hypocenter	C₅₃	#11818
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1353

368 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 69 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)(55,60)(59,67)(61,68)(62,69)(70,72,75,74)(71,77,80,79)(73,81,82,83)(76,84,78,85);(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)(58,64)(60,67)(61,69)(62,68)(63,66)(71,76,80,78)(72,74)(77,85,79,84)(81,83);(54,55)(56,62,64,69)(57,61)(58,60)(59,63,67,65)(66,68)(70,71,74,79,75,80,72,77)(73,76,81,84,82,78,83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

626 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80318);