#27300196

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12824

Subgroups

	subgroup	G/H
center	C₂²	#16810911
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	#12824
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(58,61)(59,66)(60,72)(62,75)(63,79)(64,76)(65,81)(67,83)(68,86)(69,84)(70,87)(71,77)(73,88)(74,82)(78,85)(80,89);(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)(58,60,71,80,64,73,62,63)(59,65,78,68,69,82,67,70)(61,72,77,89,76,88,75,79)(66,81,85,86,84,74,83,87);(54,55,56,57)(58,59)(60,70)(61,74,76,81)(62,78)(63,65)(64,69)(66,72,84,88)(67,71)(68,73)(75,87,77,86)(79,83,89,85)(80,82)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dh,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,i^b=i^52,e^c=egh,f^c=fh,g^c=gh

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85672);