#27290492

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#344 ⋊ C₂
Sylow	C₅₃ ⋊ #10950

Subgroups

	subgroup	G/H
center	C₄	#16810500
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂×C₅₃	#1437
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	#10950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#344

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 85 points:

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

Polycyclic presentation:

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

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, 75968);