#27304500

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1409 ⋊ C₂
Sylow	C₅₃ ⋊ #13610

Subgroups

	subgroup	G/H
center	C₂	#20276381
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#127	C₂²×D₅₃
Fitting	C₅₃×#1409	C₂
socle	C₂×C₅₃	#2127
prime radical	#16811186	C₂²
cosocle	#127×C₅₃	C₂³
hypercenter	#1409	D₅₃
hypocenter	C₅₃	#13610
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #1409

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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)(55,61)(56,66)(58,71)(59,70)(60,74)(62,79)(63,77)(64,78)(65,80)(67,81)(68,69)(72,83)(73,84)(75,82)(76,85);(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)(54,56,59,68)(55,60,64,76)(57,69,70,66)(58,72,73,67)(61,74,78,85)(62,75,63,65)(71,81,84,83)(77,80,79,82);(54,55)(56,65,67,60)(57,61)(58,63,73,62)(59,64)(66,80,81,74)(68,75,72,76)(69,82,83,85)(70,78)(71,77,84,79)

Polycyclic presentation:

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

Conjugacy classes & character table

467 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89976);