#27299164

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12608

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

462 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)(60,71,73,72)(62,76)(65,70,78,81)(67,69)(74,86,87,85)(75,77)(79,89,88,84);(58,60,62,74,63,72,77,86,64,73,76,87,61,71,75,85)(59,65,68,79,69,70,83,84,66,78,80,88,67,81,82,89);(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)(58,59)(60,70,73,81)(61,67)(62,68)(63,69)(64,66)(65,72,78,71)(74,84,87,89)(75,82)(76,80)(77,83)(79,86,88,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84640);