#27289598

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1729 ⋊ C₂
Sylow	C₅₃ ⋊ #10779

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1729	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1729	D₅₃
hypocenter	C₅₃	#10779
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2)(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)(58,61)(59,66)(60,70)(62,74)(63,78)(64,75)(65,79)(67,82)(68,83)(69,84)(71,86)(72,88)(73,76)(77,87)(80,89)(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)(54,55,56,57)(58,60)(59,65)(61,70)(62,77)(63,71)(64,72)(66,79)(67,80)(68,81)(69,76)(73,84)(74,87)(75,88)(78,86)(82,89)(83,85);(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)(58,59)(60,69)(61,73,75,82)(62,76)(63,67)(64,68)(65,71)(66,74,83,78)(70,85,88,79)(72,80)(77,81)(84,87,89,86)

Polycyclic presentation:

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

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75074);