#27349652

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17307

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator, socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Fitting	C₅₃×#1441	C₂
hypercenter	#1441	D₅₃
hypocenter	C₅₃	#17307
LCS	G ▷ C₂⁴×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ #1441

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 85 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)(56,60)(58,61)(63,67)(66,69)(71,75)(74,77)(78,80,81,85)(79,83,82,84);(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)(55,59)(58,61)(70,72,73,76)(71,74,75,77)(80,85)(83,84);(54,56,57,60)(55,58,59,61)(62,64,65,68)(63,66,67,69)(72,76)(74,77)(79,82)(80,85);(54,55,57,59)(56,58,60,61)(62,63)(64,66)(65,67)(68,69)(70,71)(72,74)(73,75)(76,77)(78,79,81,82)(80,84,85,83)

Polycyclic presentation:

2^8,53|a^2=ef,b^2=efgh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135128);