#27384359

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1841 ⋊ C₂
Sylow	C₅₃ ⋊ #20776

Subgroups

	subgroup	G/H
center	C₄	#117×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1841	C₂
socle	C₂×C₅₃	C₂×#383
prime radical	C₂³×D₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1841	D₅₃
hypocenter	C₅₃	#20776
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1841

684 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169835);