#27412637

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #23601

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133192
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×#11719945	C₂²
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	#23601
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(58,62,64,63)(59,67,68,77)(60,71,73,72)(61,75,76,78)(65,69,80,86)(66,82,84,83)(70,74,87,88)(79,85,89,81);(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)(55,57)(58,61)(59,66)(60,70)(62,75)(63,78)(64,76)(65,79)(67,82)(68,84)(69,85)(71,74)(72,88)(73,87)(77,83)(80,89)(81,86);(54,55)(56,57)(58,60,63,72,64,73,62,71)(59,65,67,69,68,80,77,86)(61,74,75,87,76,88,78,70)(66,81,83,89,84,85,82,79);(58,59,62,67,64,68,63,77)(60,69,71,80,73,86,72,65)(61,66,78,83,76,84,75,82)(70,85,88,79,87,81,74,89)

Polycyclic presentation:

2^8,53|a^2=ef,b^2=e,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^c=eh,g^c=gh,i^c=i^52

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 198113);