#27289305

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10711

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	#10711
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#344

842 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)(58,61,62,72,63,73,74,85)(59,65,67,78,66,77,79,87)(60,69,71,83,70,82,84,89)(64,75,76,86,68,80,81,88);(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)(58,60)(59,64)(61,69)(62,71)(63,70)(65,75)(66,68)(67,76)(72,83)(73,82)(74,84)(77,80)(78,86)(79,81)(85,89)(87,88);(58,59)(60,68,70,64)(61,65)(62,67)(63,66)(69,80,82,75)(71,81,84,76)(72,78)(73,77)(74,79)(83,88,89,86)(85,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74781);