#27292334

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11264

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133206
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#424	C₂
socle	C₂²×C₅₃	#400
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#424	D₅₃
hypocenter	C₅₃	#11264
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#424

578 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)(58,61,62,77,64,78,63,79)(59,66,67,84,69,85,68,75)(60,71,72,87,74,76,73,88)(65,80,70,86,82,83,81,89);(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)(58,60,62,73,64,74,63,72)(59,65,68,70,69,82,67,81)(61,76,77,87,78,71,79,88)(66,83,75,89,85,80,84,86);(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)(58,59,63,67,64,69,62,68)(60,70,72,65,74,81,73,82)(61,75,77,85,78,84,79,66)(71,83,88,89,76,80,87,86)

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 77810);