#27383737

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #20671

Subgroups

	subgroup	G/H
center	C₄	C₂×#13133173
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂×C₅₃	C₂×#380
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	#20671
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#410

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169213);