#27383950

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1740 ⋊ C₂
Sylow	C₅₃ ⋊ #20710

Subgroups

	subgroup	G/H
center	C₄	#16810485
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂×C₅₃	C₂×#381
prime radical	C₂²×#11719945	C₂²
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#20710
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #1740

1276 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)(54,58,59,71)(55,62,63,75)(56,66,67,80)(57,69,70,82)(60,73,64,77)(61,74,68,81)(65,78,79,85)(72,83,76,84);(54,57)(55,61,63,68)(56,65)(58,69)(59,70)(60,72,64,76)(62,74,75,81)(66,78)(67,79)(71,82)(73,83,77,84)(80,85);(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)(55,60)(57,65)(58,66)(59,67)(61,72)(62,73)(63,64)(68,76)(69,78)(70,79)(71,80)(74,83)(75,77)(81,84)(82,85);(54,55)(56,64,67,60)(57,68)(58,62)(59,63)(61,70)(65,72,79,76)(66,77,80,73)(69,81)(71,75)(74,82)(78,83,85,84)

Polycyclic presentation:

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

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, 169426);