#27603823

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2811 ⋊ C₂
Sylow	C₅₃ ⋊ #37194

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2811	C₂
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2811	D₅₃
hypocenter	C₅₃	#37194
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2811

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 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)(71,77)(75,78)(76,81)(84,85);(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)(58,60)(70,73,74,83)(71,76,77,81)(72,79,80,82)(75,84,78,85);(56,61)(58,60)(62,64,65,69)(63,66,67,68);(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)(54,56)(55,58)(57,61)(59,60)(70,72,74,80)(71,75,77,78)(73,82,83,79)(76,85,81,84);(54,55)(56,60)(57,59)(58,61)(62,63,65,67)(64,68,69,66)(70,71)(72,78)(73,81)(74,77)(75,80)(76,83)(79,84)(82,85)

Polycyclic presentation:

2^8,53|a^2=g,b^2=fh,c^2=g,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,i^b=i^52,i^d=i^52,i^e=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 389299);