#27300445

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12881

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12881
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

298 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)(54,56,57,55)(59,66,67,68)(62,77)(65,79,70,80)(69,84,83,82)(72,76)(75,78)(81,89,85,88)(86,87);(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)(54,55,57,56)(58,60,61,71,64,74,63,73)(59,65,68,80,67,70,66,79)(62,76,75,87,77,72,78,86)(69,85,84,88,83,81,82,89);(55,56)(58,59,62,69,63,66,78,82,64,67,77,83,61,68,75,84)(60,70,72,85,73,80,87,89,74,65,76,81,71,79,86,88)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,i^b=i^52,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, 85921);