#27429181

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1945 ⋊ C₂
Sylow	C₅₃ ⋊ #24985

Subgroups

	subgroup	G/H
center	C₄	#16811339
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1945	C₂
socle	C₂×C₅₃	#2420
prime radical	#20284436	C₂
hypercenter	#1945	D₅₃
hypocenter	C₅₃	#24985
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1945

646 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)(54,58,59,71,61,70,60,56)(55,64,65,62,67,68,66,79)(57,74,75,85,77,73,76,69)(63,81,72,78,82,80,83,84);(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)(55,63)(56,69)(58,74)(59,75)(60,76)(61,77)(62,78)(64,81)(65,72)(66,83)(67,82)(68,80)(70,73)(71,85)(79,84);(54,56,60,70,61,71,59,58)(55,62,66,64,67,79,65,68)(57,73,75,69,77,74,76,85)(63,80,72,84,82,81,83,78);(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)(56,68)(57,72)(58,62)(59,66)(60,65)(61,67)(63,75)(64,71)(69,84)(70,79)(73,81)(74,80)(76,82)(77,83)(78,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 214657);