#27301228

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13023

Subgroups

	subgroup	G/H
center	C₂²	#16810755
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#510	C₂
socle	C₂²×C₅₃	#447
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#510	D₅₃
hypocenter	C₅₃	#13023
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#510

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,i^a=i^52,d^b=dg,e^b=egh,f^b=fg,g^b=gh,i^b=i^52,d^c=dgh,e^c=egh,f^c=fgh,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, 86704);