#27271035

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7180

Subgroups

	subgroup	G/H
center	C₈	#13133148
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂×C₅₃	#1279
prime radical	C₂³×C₅₃	#104
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#7180
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(54,56)(55,62,65,78,67,80,66,79)(57,69)(58,77)(59,71)(60,72)(61,73)(63,68,81,83,64,82,74,84)(70,75)(76,85);(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)(54,55,58,64,60,66,76,81,61,67,75,63,59,65,57,74)(56,68,70,79,72,84,69,80,73,82,77,78,71,83,85,62)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,i^a=i^52,d^b=dh,i^b=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, 56511);