#27349758

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₈×M₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #17328

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#17328
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×M₁₆

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 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)(72,78)(75,81)(79,84)(82,85);(55,60)(59,65)(61,67)(66,69)(70,72)(71,75)(73,79)(74,78)(76,82)(77,81)(80,84)(83,85);(54,56,57,62,58,63,64,68)(55,59,61,66,60,65,67,69);(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)(56,59)(57,61)(58,60)(62,66)(63,65)(64,67)(68,69)(70,71,73,76,74,77,80,83)(72,75,79,82,78,81,84,85)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,e^2=g,f^2=h,c^a=ch,i^a=i^52,d^c=dg,i^d=i^52

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135234);