#27271272

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7271

Subgroups

	subgroup	G/H
center	C₂²	#16810371
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#333
prime radical	C₂³×C₅₃	#100
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#7271
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(58,60,61,71,74,72,87,86,62,70,75,85,63,69,76,84)(59,64,66,79,77,68,82,83,65,78,80,89,67,81,73,88);(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)(58,59)(60,68,70,81)(61,73)(62,65)(63,77)(64,69,78,72)(66,76)(67,74)(71,79,85,89)(75,82)(80,87)(83,86,88,84)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=fh,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^52,c^b=cg,d^b=dh,f^b=fh,g^b=gh,i^b=i^52

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56748);